What did you end up doing? If the home is sinking, on land that is not the home owners, do you have a clause in the rental agreement that states everything above the ground is the tenants responsibility, apart from water / power attachments into the ground? If not, hope you talked to your lawyer first.
We had a home start to sink last winter and it turned out an underground vault collapsed and took the end of the house with it.
We didn't know it was there until we started digging around under the corner where a sink hole appeared. But, we put a document together with the lawyer that said we'd jack the house and level it but any and all repairs to the home were the home owners responsibility.