How to locate a cut low-voltage ethernet wire behind drywall?
You need a LAN Tracer



This one is from Amazon - https://www.amazon.com/Upgraded-VicTsing-Ethernet-Telephone-Tracking/dp/B008G8KE90

Connect the sender to one end of the cable, and use the tone receiver to track the cable in the wall. More expensive tracers can tell you the length of the cable up to the break - but I prefer the tone, which tells you exactly where the break is. 

Trace it from both ends, and you should end up in the ballpark so one hole will suffice.