I am letting a person use a room in my house that was kicked out by his girlfriend. It was a mistake and I want him to leave. The police say that they will not force him to leave although he has not given me money or has a lease.
What are my rights in Texas?

How long has he been there? Probably long enough to establish residence, which is why the police won't force him to leave. So he's no longer a guest, but a resident and has to be treated like a tenant. This means you have to give him 30 days written notice to vacate. If he still doesn't leave, then you will have to go to court for an eviction order.

He has been here 2 weeks. Does that make a difference? He has not paid any rent or received any mail.

He has been here 2 weeks. Does that make a difference? He has not paid any rent or received any mail.

Only 2 weeks? If the police won't treat him as a trespasser, then you will need to evict him