Trouble with Toddler Girlfriend

Honestly, I can't stand when other moms give advice that would benefit the other mom, the mom who isn't asking the question, so since you are asking the question, and you are looking for what makes the most sense to you, I would agree with some moms that say be honest, but we all know how hard that can be, and it may create a problem in the future. It is not your fault that her daughter doesn't want to play, and it is not your fault that maybe the other mom doesn't have many friends. You can't be a charity case for everyone, there just isn't enough time or energy in life for that, plus it isn't fair to either of you to have a fake friendship. So, reading your post, it sounds like you want permission to not talk with her, here I give you permission- Just say you are super busy with two kids, it is a big transition, you are not ready to take going out with two kids on your own, you are busy, tired, sleep deprived, etc. If you want to get together down the road, the option is still there, you never said anything to hurt her, and as time goes on, you can decide if it is worth it. Congrats on baby #2 and remember to put you, your daughter and family first. I hate listening to other moms saying you should go and try it again and again and again, why? Again, there are so many people in this world, you could be lovely aquantances, lol :)

I would ask her how the girl is doing with playdates. I have a friend whose son hated Meet Ups when there were more than 2 kids because he doesn't like crowds. Does her daughter do better in her home? Would you be willing to go there? Or do you have a library you could visit together where if the girl is quiet there might be other kids or you could read to your DD or attend a storytime? You could also tell the mom that you are at a loss because while you enjoy HER company, it seems the girls don't do so well with such different personalities. Does she have suggestions? You can say you haven't known what to say about it which is why you were slow to respond.

Sounds like the mom wants your company more than her daughter wants your daughters company. Maybe she uses her daughter as an excuse just to visit you and have some adult conversation. You might be the only friend she has since her daughter is so shy and maybe other moms have already stopped playdates with her.
I think you should be honest with her and let he know that your daughter always asks why her daughter wont play and at this juncture you don't want your daughter dealing with that disappointment anymore. Your poor little girl probably gets pretty confused by this.

added: i also think if you enjoy the other mom go to places they don't have to play together, indoor playground...u would get a break b/c they would play and there may be other kids too
i would ask the mom to talk to the girl and see if your daughter did somthing at one point to hurt her feelins and thats why shes standoffish. This way you're not accyusing her kid of being wierd but addressing the problem. My daughter is the same way you describe her daughter to be and I know when she plays with kids that aren't as quiet and sensitive sometimes the typical toddler problem "susie pushed M." or "susie called M. a name" can hurt these shy kids and make them hesitant on continuing to play. I'd ask the mom to see if she can find out what the issue is, also maybe do it at one of your homes, so if they don't want to play together they can play on their own...at this age most kids still play on their own anway

I dont think the other child not playing is a reason to put a stop to the playdates, if you enjoy the other mom's company, and if you feel up to going.

Pick someplace where your daughter has fun no matter what and have the play date there. If the other girl plays, good, if not, your daughter can play by herself. Explain to your daughter before hand that the other girl may be too shy to play - to continue to try to be a good friend and play, but that she may play by herself. I dont think you need to rescue the playdate by playing with your daughter.

However, if you just dont want to have a playdate for whatever reason, tell the mom no. You have a newborn, and its hard and your are tired, and just not up to it.

I dont think you should got into any explanations about her daughter not playing - that would sound like you are criticizing her child. Its totally fine that she is shy.

I really think I would just tell her that it is more difficult to have playdates when you have a tiny one. And leave it at that.

Could you try a playdate where the girls are alone together to see if they'll play when the moms are involved? Maybe you could call and ask if you could swap "babysitting" - she could drop her daughter off for a few hours sometime, and you could then drop your daughter off at her house for a few hours sometime.