The legislative branch can pass a law requiring enforcement, likely within some specified parameters or timeline. If that passes and is constitutional, the courts could be tested and uphold the law.
Ok, but (a) they won't, and (b) if they do, who carries out the actual removal of the president from power? ... oh, right, the executive branch, again. Oops.
If your concerns are only procedural, surely congress could fix that if they cared. If they actually had the vote to impeach they could likely have the voter to either pass new law or amend the constitution to ensure the removal is enforced.