Internet over Ethernet cables connected to a phone key system

I'm trying to setup several wired connections in different rooms in a large house (~5000 sq ft) for a family friend.

Currently, it looks like when they originally built the house they ran both Ethernet and coaxial cables to practically every room. However, in each room the Ethernet cable is connected to an RJ11 keystone jack rather than an RJ45 one, and it is only using 1 or 2 of the twisted pairs. On the other end (in the basement), the Ethernet cables are punched into a patch panel, which connects to a NEC DSX-40 phone key system.

What is the best way to get wired internet connections to these rooms? Ideally we want to also keep the phone line extensions available for each room.

-Someone suggested using MoCa adapters, but that seems expensive and overkill since there are already Ethernet cables running to all the rooms. - I read an article about using a single Ethernet cable for both phone and internet, but would that work with the phone key system? - If we can't reuse those Ethernet cables, is it possible to use them as guides for fishing an additional cable?

Thanks in advance for any advice!