RE: XPS 15 9560 Killer Wireless killing my network
I am with Rivet Networks, the company who produces and supports the Killer Wireless 1535. Unfortunately your issue seems to lie with the Wireless Driver in the router. If the Wireless is stopping for the entire router, it sounds like the wireless driver in the router is crashing. Have you looked to see if there is a firmware update for your router?
I might also recommend to try other Channels, Frequencies (Mhz), or wireless mode settings to see if that can help.