This is a tough decision to make. And I don't see much in the way of positive options.
As for how mama mouse got in, mice can squeeze through any opening the size of their skulls, which is extremely tiny, about the size of a nickel. She even possibly got in from another location in the house.
Unfortunately, without a mother, the babies have an almost nil survival rate. It's very /very/ difficult to hand rear pinky mice, even by experienced mouse breeders. Whats more, mother mice are notoriously skittish and pragmatic about their offspring. If they are stressed and disturbed, they will often abandon or cannibalize their young in an effort to cut their losses, recoup lost calories, and hopefully have a new litter in a safer place.
You can try livetrapping the mother, but I'll be honest, I'm not expecting much success. If you have an existing pet mouse mother, she might be able to foster for you.
Your current options are indeed, to remove your mouse infestation by hand and hope that was the only litter. You can humanely euthanize them or try to hand rear the babies to be released elsewhere. Your other option is to leave them to be raised by their mother and have a colony established in your house which will reproduce quickly and possibly carry in diseases, and try to discourage or eradicate the colony yourself.
Here are some links that will hopefully help you:
Info on dealing with infestations: http://www.cdc.gov/Features/Rodents/
Fostering Baby Mice: http://www.thefunmouse.com/info/orphanedmice.cfm
Best of luck to you. I know this isn't the advice you were hoping for, and I know this situation sucks. I think it's awesome your heart wants you to help them, and with luck and work you might succeed.