If you think you're a victim of ID theft, you should first contact your local police department and file a police report. They will be in the best position to go over the facts with you and will decide if criminal charges will be pursued.
With respect to your landlord, much more information is needed here. I would suggest you contact an consumer advocate in your area who is seasoned in fair housing and landlord-tenant issues. Go to www.naca.net, which has an attorney search feature, to find an advocate in your area. Many attorneys will undertake an initial review of your matter without cost of obligation. He/she will be in the best position to give you information about your legal options.
All the best to you with your issue, and your health issues.
NOTE: This Answer does not constitute legal advice. Every case is fact specific. To render a legal opinion, an attorney must engage in a consultation with a prospective client and review any pertinent documents. This communication does not create an attorney-client relationship with Attorney Amy L. Wells or WELLS LAW OFFICE, INC.
The other lawyer has given you good information. If you are not doing something illegal like working under the table then Social Security will not have any reason to investigate your situation.
This response is meant to be information only and should not be considered to be legal advice. This information is not meant and should not be construed to be the formation of an attorney client relationship. I practice Virginia Workers compensation law and Social Security Disability law.