Getting the backing paper off laser cut acrylic is a pain. Some people recommend d-limonene, a citrus-derived solvent that is the main active ingredient of Goo Gone.
Pour a little Goo Gone into a tough freezer baggie, and place your acrylic part inside. Seal it up, and lay it flat for a few minutes. After that, flip it over and let the other side soak for a few. Open the bag and fish out your work. The backing paper should just slough off. Now rinse off the acrylic with washing-up liquid/dish detergent and warm water, taking care not to scrape the surface. You should now have a perfectly clean and shiny acrylic object. The d-limonene has the pleasant side-effect of de-stinkifying the cut plastic, too.
You should be able to re-use the Goo Gone baggie many times if you’re careful. You might not be able to rinse Goo Gone down the drain where you are; please check local regulations.
(The piece is the non-broken version of this.)