I’m a nanny for a rich family that lives out of state. I ran into the Dad at a bar. One thing led to another and we had sex. I’m not that kind of girl and I feel awful. Do I tell his wife and lose good income or pretend like it never happened?

How could you continue working there, regardless of what you decide to do, I don't see it as an option, at the very least it would be extremely weird and uncomfortable. Get out of there.

You kinda chose to be that girl one night that we know of. It was a decision you made. Guilt is like a grain of sand in your bed. I honestly would exit the situation permanently without a word of the real reason, and find another new job that pays great, and where you are never that kind of girl again. Its a do over !

Being 'that kind of girl' is irrelevant, It doesn't matter if you have never done this before, the fact of the matter is .. You (and he) DID something you shouldn't have, that will offer consequences .. To not only your future, but an entire family. 

I'm not sure what's worse .. Fessing up, and allowing the truth to tear this family apart .. Or .. Going to work, and facing the woman who trusts enough to allow you to care for her children, and continuing to deceive her to her face, everyday knowing what you did. 

This may be a little harsh but .. You can't 'undo' what you've done.  If you have any decency, find another line of work.

As a woman and mother, I can tell you .. If it was me, I would want my husband to be revealed for the piece of crap he is. But that's just me.

You work for a dog! Isn't there a shelter you can volunteer your self to! At least there you will be dealing with the four legged types! FIND A NEW JOB AND DON'T BE SELFISH SAYING "OH I AM LEAVING SOOO MUCH MONEY!" Pfft!