Use cat5e 2-pair RJ11 cabes for network connection
The best solution you can arrange for 2-pair cables will be limited to 100 Mbits/second (or 10 times slower than the current most common standard wired ethernet speed.)

Rather than looking for an adapter (dubious at best - any you find will likely be intended for telephone signals rather than data), I'd suggest just removing the RJ-11 jacks, and replacing them with Cat5e RJ-45 jacks. 

You'll connect your two pairs to the "Orange &amp; White/orange and Green &amp; White/green" marked terminals (1,2,3,6) and ignore the blue/brown marked terminals (4,5,7,8)

Edit: Since the cables have been discovered to be normal 4 pair cables after all, connect all 4 pairs, and Gigabit (1000 Mbits/sec) speeds should be possible across them.