Sometimes my one and a half year old daughter wakes up in the morning before I have a chance to daven shacharis. My wife is still sleeping at the time, so I don’t want my daughter to cry or go and wake her up. I watch my daughter for some time until I go to daven. Is it permissible for me to read her stories, play games with her, etc. to keep my daughter occupied and calm, or might engaging in these activities be problematic before davening?
It is permitted. Although we should not engage in work before davening, however work that is mitzva related is permitted. Helping one’s wife and caring for his children are a mitzva, therefore it is permitted.
You should have a lot of nachas from her.
Halichos Shlomo (Tefillah) 2 ftnt. 8