Can we legally ask her to leave.
Yes, you can ask her to leave.
She, on the other hand, can simply say, "No, I ain't leaving because living with you is FREE for me."
If she leaves, all the better for you.
If she refuses, you'll have to file an eviction action against her through the applicable local court.
Chalk this up to one of life's lessons and never allow anyone but you, your spouse, and minor children to live under your roof.
Can I change my locks and alarm code.
Yes, you can change your locks and alarm code, but NOT until she voluntarily leaves (which we BOTH know will never happen), or until the court FORMALLY evicts her.
There might not be a written lease, but your state legislators have remedied that for you by creating what some call a "month to month" tenancy.
Don't try any shortcuts, or you'll only LOSE more money!