Can I draft a will that will leave my girlfriend (not spouse/domestic partner) my house that has a remaining balance on the mortgage?

Yes, you can leave her the house, but there will still be a lien on the house for the amount of the unpaid mortgage loan. In the event you pass, the bank will assert the debt against your estate. She has the option of taking on the mortgage if she wants to continue living in the house, or selling the house and paying off the mortgage with the proceeds.