emsr2d2 gave the answer I should have gone..., which means the correct room I needed to go to; while my question is whether I should say the wrong room I went to should be/should have been XYZ.
Did you mean that you wanted to use the word "should" in the same part of the sentence as the wrong room? If that is the case, then you need the negative:
I should not have gone to Room 110. I should have gone to Room 101.
I went to Room 101. I should not have gone to that room.
Remember - if you don't use correct capitalisation, punctuation and spacing, anything you write will be incorrect.