I can't speak to the M&P 22 other than to say, I want one!
I can say that the SIG Mosquito seems hit or miss. I own two, one purchased in 2009 that runs OKish with CCI mini-mags and federal bulk. The other purchased in 2011 will only run on mini-mags with either recoil spring. Also the sheer number of threads regarding the problematic Mosquito should have prevented me from investing in a second one. I'm less than satisfied with both of the guns. None of my other semi-auto 22s require the level of patience that the mosquito has required. Break-In periods, Jams, failures to feed/eject/extract, out of battery discharges... the Mosquitos have shown me nearly every semi-auto failure you can have. Not to mention SIG's customer service... hope you enjoy being ignored and/or playing phone tag.
Geez... I did not intent to be this critical; I'm a huge SIG collector. They do make quality products but I just can't like or recommend the Mosquito. I can however attest to the quality of the full sized guns with the 22LR slides. I have a P229 and P220 with caliber exchange kits. I don't have any problems with them and the shoot whatever brand of ammo is on sale. The lack of last shot slide lock is mildly annoying though. Probably too big for your daughter's hands though.
These are opinions, based on my experiences. YMMV.
