It sounds like the salon owner doesn't own the lease.. She must, just, be renting the salon from the landlord, who owns the lease.
It is common practice for a landlord to not allow sub-letting or renting, which unfortunately, is where you have come unstuck.
Although the salon owner has broken the landlords rule not to sub-let, I don't think she's done anything wrong by asking you for rent, as long as you've used the space and facilities.
Hopefully the landlord will let you pay your rent directly to him/her and allow you to continue.