There are nuances of difference in the constructions about to whom she was being insistent -- to the world in general or to you in particular (in using "to me" -- and whether the focus was on getting her room cleaned or having you and no one else do it. If you say "She insisted that I should clean her room", it's presumably you who should do it because you're the one who is around. If you round up some other sucker to do it, she may not care so long as the room gets cleaned. If you say "She insisted on me cleaning the room", then it's definitely you she wants, as opposed to anyone else available.

She insisted that I (should) clean her room.

She insisted that I clean her room.

Thanks for your help ,Beytullah

I think it should be something like this: She insisted that I should clean her room.