Her brother should have told her that,

1. A verbal contract counts in the absence of written one.
2. Written contact has more weight as it proves the agreement. Every lawyer I know drills this into their friend and family. Mine son even wrote mine because it that important.
3. You can not create conditions of contract after the event. My son explained this as meaning just because when you made this arrangement, you didn't discuss what would happen if you left the day care/babysitter early, that does not mean you can automatically demand a refund unless that was discussed and promised as term of her providing care.
4. Full circle situation is, if you did discuss it and it was part of terms, now how can you prove without written contract? Which is why, verbal only work when both people in agreement.
5. You cancelled care due to your change in situation - quitting job. That not mean you can pass your changed financial situation onto someone else to deal with. Of course you entitled quit job but there is consequence to that. Just like if you had bought monthly bus pass to get into work, there is no refund for unused part you decided you needed at start of month.
6. I think you using the term babysitter to diminish her achievement and to minimize her role. That will bite you hard. If she really babysitter, that might mean she likely your employee. And if she your employee, then Labour Standard Rules might apply. If she just casual babysitter, then most don't have to give receipt. No teenage baby sitter write one up at end of the parent's monthly date evening.
7. If she day home, you should get tax receipt but she not have to give before end of Feb. like all other places have same time for tax related documents to be given out.
8. For $500, it sound like you negotiated a cash price. That is illegal to work under counter but you were part of the illegal deal too. A cash price will not give you receipt. And if you go to court, it likely that Judge won't support your claim for either refund or receipt because you are just as guilty as her for striking illegal arrangement. I think my son said it is known as having "clean hands". You not able to sue someone for their part in illegal transaction when you just as guilty. For instance, drug dealer couldn't sue customer who paid with forcery money because court must not reward him for selling illegal substances.
9. Basically, based on information you say, you are out of gas. As someone who claim to be legal secretary you should know how important it is to get everything in written. As someone whose brother is lawyer, then you likely have heard this million times from him.
10. Most answers here from day care providers. It 100% unanimous that no refund is due which very rare from what I seen here. Since we all agree, it likely that it very black and white situation. I hope your provider is reading this thread and experienced enough to stands her ground and not be hassled into backing down.