Should I go through her stuff secretly? What if she catches me looking through her stuff, can she report me to the police? What should I do?
The biggest challenge is you have no idea what she might be stealing or if she is stealing at all. The first thing to do is establish whether she really is stealing. Just asking her is one option, but if it turns out your helper is not actually stealing, then it will probably sour the relationship.
Ideally is to always know exactly how much is in your wallet and what valuables you leave lying around. Always keep money or valuables out of your helper’s sight. Most helpers are not thieves, but even for an honest person (helper or not) the temptation to steal cash or jewelry openly visible left on, say, a kitchen counter, may be too much. Before you hire a helper, get a small safe and keep valuables in it. Also keep cash out of sight at the very least.
So if you are sure something is missing, check with your family member if anyone has taken the money or an item before accusing a helper. You can also try to come home at different times than usual or somehow changing your routine. Perhaps you can catch her in the act.
While technically you can enter the helper’s room, you should not search her things. It is a violation of her privacy. She should be treated as innocent until proven guilty. If you are certain she is stealing from you, do not act on your own. If you do so, it will be your word against hers. Have a friend or your partner present so that you have at least one witness when you confront her with your suspicions. If she has been stealing objects and not cash, ask her if you may search her room. If she refuses, you can always call the police. Unless the sums involved are large, I would try to refrain from using the police since it adds a lot of complications like paperwork and the filing of formal complaints. If she has been stealing cash, this is harder to prove. You could resort to things like marking banknotes or writing down the serial numbers, but this seems a bit excessive.
Once the helper starts working, build a rapport with her so that she feels loyalty towards you beyond just the paycheck, and is thus less likely to steal from you.