Can I sue a doctor for medical malpractice ?

You can do both. The landlord is responsible for the doctor's malpractice. Your lawyer should know this.

You can sue the landlord as well as the doctor.

While you can proceed separately, I would bring the two actions together as in my view, it would be beneficial to your case when it comes time to a trial on your damages. Not only that, if the cases are brought separately and you claim the surgery and revision in the slip and fall case (which you should), there could be an offset of damages in a separate malpractice case. If the cases are brought as one, everything would be decided at once and you could very well have a situation where the defendants from each incident are fighting amongst themselves as to the cause and severity of your injury in an effort to limit their own exposure. Thus, you could have the added benefit of the defendant in the slip and fall claim blaming the surgeon for making your injury (and damages) worse, and the surgeon testifying that your injury was so bad when you went to him that they did the best they could under the circumstances. This would all be to your benefit in my view. If the cases are brought separate, you lose that benefit. I would have an in depth discussion with your attorney about this and listen to the pros and cons of proceeding either way so that you can make an informed decision. Good luck.