Give it time, she is young and probably feels like you want to replace her dad. Don't try to act like a father to her but be like a friend. Tell her you want to be her friend. I think its sweet of you that as a step dad you bath, feed and play with the child, some bio dads don't even do that they leave it to the mother. With time she will learn to appreciate all you do for her, but you need to know that she is not your child and she knows that, she actually has a father she loves so yes you will always come third after her mom and dad, thats just normal and natural, don't be bothered by it. All you can expect from her is respect, appreciation and friendliness. 

Here is the good, the bad and the ugly of it all.

The good, she is young and you have a lot of time to spend with her.  You may have a decent opportunity to bond with her and develop a close relationship. 

The bad... well, you aren't her father and well, your relationship may never get to that full on parental point.  But it still can be good and she can still look up to you, learn from you and yes even love you... but if she at least likes you, that's ok too.

The ugly... It's quite likely you will always at the end of the day be third in line for her attention/affection.  She has 2 bio parents.  Even if you spend more time with her than dad, if you were both in a burning building, she probably saves him first.  On the one hand, that really sucks.  But, it's nature and a fact you will need to wrap your head around.  Does that mean that you shouldn't even try or care? nope.. we care about people all the time without expecting a tit for tat response.  The ugly is that even after a lifetime of caring for her she might turn her back on you for her parents.. or if you split with mom.. you get forgotten.

In the end, try to focus on the positive and be a supportive partner to your wife.  Get to know the daughter... help care for her if you want...teach her things and love her as much as you want to...  But, keep in mind that her parents are also there for her and that your position may be more a supporting role than the lead.

Give it time.  She doesn't need to love you, and you don't need to love her, it isn't really fair for your fiance to ask that you do.  Love is something that comes with time, or it doesn't.  A lot of bio parents have unrealistic expectations about their new partner bonding with their kids.  You are obviously being a really good step parent and involving yourself in your SD's care. Try and remind yourself that she is only 2 and maybe is having difficulties coming to terms with the changed family dynamics - it can take a few years to settle.  The big advantage is her tender age, and as long as you do not become resentful of her and reject her, she is unlikely to reject you.  Try not to take it to heart that she acts out sometimes, she's just a baby. 

didn't speak to my husband for almost two years after his second deployment unless she had to and he is her biological father. She was two when he left and almost five by the time he was home for good. Even when he came home in between deployments and such she barely acknowledged him or cried. She didn't know him and it is very common for young kids to have a favorite even between their parents. Just keep doing what you are doing and let things happen naturally. Your fiance needs to do the same. 

Also, kids that age do not understand appreciation or empathy. It is too conceptual and their young brains aren't there yet. They really just expect someone to bathe, clothe, feed and play with them. This is also the age of separation anxiety, routine, and tantrums. This child's routine and family are changing plus she sees daddy very little. 

Welcome to ST.  There are a few of us who have been Step parents for 10 15 nearly 20 years. Most of us dont sugar coat stuff.

First things first, you are not a step parent. You are the live in boyfriend with plans on getting married. Whens the date? Congrats !!

You really seem like a nice guy who is trying to figure out where the He** you fit in with allllll this. We get it.

Next....DO NOT give this child a bath. That is the mothers duty (and bio dad) where on earth is BM that you are generally taking care of her 5 year old. Your words not mine.  I am sorry it sounds harsh,, it is harsh and that is the point. 

The best way to describe a role of a step parent (which your not yet) IS, similar to an Uncle or Aunt. Your kind and  Your the fun guy....but not the bather or shower giver, not the homework person, not the bank ATM, not the disciplinarian either or transportation CEO. Your going to be a spouse of a woman who has a child.  SHE is the parent not you. Bio Dad is the parent , not you.  

You are not here to piggy back off of her Father. YOU are going to be the husband of your wife. NOT a 3rd parent.

Why does bio dad ONLY have 2 days a week. Is that Friday - Sunday? His daughter deserves more time with her dad than that. That a darn shame. Little girl should have no less than 50percent with each bio parents. Unless he was charged with child abuse or child neglect.

Look at everything you are currently doing. Ask yourself...why am I doing this. WHAT would my girlfriend do if I was not in this picture. Who would take care of her daughter? Who would give her 5 year old daughter a bath?

I strongly suggest you stop that today.

My steps are older. But my I can remember my bio son telling me when he was really little - repeatedly - I was fired and that I should go to another family. It's kind of a kid thing - it sucks. It does hurt. Shake it off if you can.

One thing I can tell you about step parenting - in my limited experience of a year - is that even if everything is going great - there is still doubt in your head. Do they really like me? Will they forgive me if I screw up in a parenting moment (and you will, we all do). Why do I find this kid so annoying? (Because kids are annoying, well - humans are annoying). I am lucky because I have a BS about the same age as my steps. So every little annoying thing my steps do, I can compare to an annoying thing my son does and then I let it go. 

One reason our biological parental bond is so strong is because if it weren't we'd abandon the little blighters - or worse. Kids are hard work. You give and give and give - and it's only the love you feel that gives you the reward. You can't expect consistent emotional rewards from the kids themselves because they aren't able to appreciate or be thankful or empathise or understand how hurtful their words can be - although yes, there are rewards a-plenty. 

I approached my step parenting relationship from a philosophical perspective- if it's love that makes us overlook all the drama - then I'll choose to love my stepkids. Is it the same love I feel for my son?- no. Is it similar? yes. I dont' expect my partner to love my son the way he loves his kids. But both of us treat each set of kids as if we do. It works for us. 

My wife and I met and started dating when my SS-26 was 15mos old.  In short order I came to clarity that my bride was the woman I wanted to make my life with and to make that  happen I was going to have to be Dad(dy) to my SS.

This was not an easy thing.  I had what I can only describe as a mammalian visceral refulsion to the presence of another man's child in my life at that early stage of the relationship. Don't get me wrong. It was not a constant thing. I enjoyed the kid, we had a great time together, and I cared very much for him but periodially it was a like a switch flipped and I had a gut wrenching reaction to his presence.  In our case it was all me and not anything he did.  He was a toddler and entirely innocent.

I equate this to an Animal Planet special on lions I once watched. When a new male takes over a pride the first thing that usually happens is that he kills the young offspring of his predicessor so make room for his own progeny.  Fortunately... we are humans and not lions and as such we can make different decisions.

So, to build the feeling of love for the kid I took the actions of love. I carried him on my shoulders, his mom and I would swing him between us when we walked, he and I chased ducks around the golf course together... it did not take long for the feelings of loving him as my own began to grow as a result of the actions of love.

Meanwhile 24 years later and back at the Ranch... our son just had his 26th birthday.  His mom and I are very proud of the man that he is and for his service to our country (Seven+ years and counting in the USAF).  In 2015 he asked me to adopt him (he was 22) and we made that happen. Four days after we contacted out attorney the Judge signed the adult adoption order.  That was one of my proudest days let me tell  you.

So, deep breaths, take the actions of love and the feelings will grow.  Kids are smart. They know who has their back, who loves them and who their REAL parents are. And that does not have to have anything to do with biology.