I have the same laptop. I wanted to run Debian/sid on it so I knew going in that I was going to need to do something about the wireless card. I researched[1] a bit and bought an Intel Wireless 7260 for ~$25, swapped it in, and it made all the difference.
Thanks. The proprietary Broadcom driver worked well enough until it didn't, and now I know firsthand why not to use them, so that's a useful experience at least.
The card identifies itself as:
[1] https://yurovsky.github.io/2015/05/18/dell-xps-13-wifi-linux...