did I broken my screen trying to replace the battery

Hi Charles,

Unfortunately, it does sound like you damaged the display during the repair process.

Fortunately, however, Ifixit does sell replacement displays for the 12 mini. A link to it and the parts required for the repair can be found here: iPhone 12 mini Screen: LCD/OLED + Digitizer Repair Part, Kit (ifixit.com) Note you can select the LCD option instead f the OLED option which whilst it is cheaper, will give you lower image quality.

Otherwise, there are cheaper displays such as this one: OLED/INCELL For iPhone X XR XS Max 11 12 Pro Max Mini LCD Screen Replacement Lot on eBay, however these will be of far lower quality that the Ifixit replacement. If you let me know your country of residence, I can find you a higher quality replacement that should be cheaper.

I hope this helps, let me know if you have any other questions :)