Your logo could be copyrighted or trademarked, depending on what you want to protect: artistic creation or identification of the source of a product/service. If someone is already using your name (as opposed to logo) this is beyond a do it yourself project in terms of a cease and desist letter and or registration. Perhaps you and the kidney center can collaborate for your mutual benefit.
The above is general legal and business analysis. It is not "legal advice" but analysis, and different lawyers may analyse this matter differently, especially if there are additional facts not reflected in the question. I am not your attorney until retained by a written retainer agreement signed by both of us. I am only licensed in California. See also avvo.com terms and conditions item 9, incorporated as if it was reprinted here.
I think you'd do well to talk to an attorney about this issue. As the other attorney has mentioned, you are beyond the point where applying for copyright or trademark protection is a do it yourself process.
This information is for educational purposes only and is not intended to form a client/lawyer relationship. I currently work for North Carolina Prisoner Legal Services, Inc., a non-profit law firm which exists to protect the rights of prisoners in the custody of the state of North Carolina. If a loved one is in prison in North Carolina, advise the inmate that they may write to North Carolina Prisoner Legal Services, Inc. NCPLS will review their case at no cost and will litigate at no cost to the inmate if the case meets NCPLS' standards. NCPLS can also provide some assistance to inmates seeking to represent themselves. I am also available for cases that do not deal with inmates incarcerated in North Carolina's jails and prisons.