A college student or single mother would probably gladly accept an offer of reduced (or free) room & board in exchange for reasonable amounts of household help. If you're expecting a full-time employee, you'll probably need to offer a small wage on top of room & board. And most exchange student programs would not allow you to use a student in that way, as far as I know.
I'd try posting on craigslist - investigate the value of room & board in your area, and then offer a STEEP discount for a set number of hours of services. For example, if your room would rent for $300/mo, and you're expecting 15 hours of child care from them per week, offer the room for free and try to negotiate a contribution towards household groceries (like $100, maybe?). At that exchange, they're "working" for $5/hr ... so just keep that in mind when proposing your deal. =)
If you're not intending to have "set" work hours (like, you need someone to watch the kids but you can't say you'll always need help Mon-Weds from 3-8p), I'd estimate the hours you might need someone and then see what you can find that might suit. It would be hard to have someone essentially on-call all the time, so I'm not sure how those negotiations would look, tbh.