Information for the 13 mini seems to be in short supply. I found one video that shows the final stages of removal, assuming you've already taken out everything in the way; in particular the rear cameras, so you'll have to do some homework to figure out how to get there such as any kind of teardown videos available. Here's the video.
[link|https://vimeo.com/630066923|iphone 13 mini flash removal- on Vimeo|new_window=true]
Looks like the part is available; I usually start with Amazon, eBay and AliExpress, but here's one I ran across just to get you started.
[link|https://www.unionrepair.com/replacement-for-iphone-13-mini-flash-light-flex-cable.html|Replacement for iPhone 13 Mini Flash Light Flex Cable|new_window=true]
Hope that's what you were looking for; let us know how it goes!