There are a few ways to approach this:
Make a mechanical device to actually physically disconnect and RJ45. Cut the tab off of the RJ45 plug, static mount an rj45 coupling and solenoid to a board, and use the solenoid to pull the rj45 out of the coupling. Might need some guides or rail so the rj45 goes back in readily. This replicates a connector being ...
Many network PHY chips can measure the flight time of a pulse. When a sent pulse is reflected back at the open end and the echo arrives back, the time it took can be used as an approximate estimate of the cable length.
It is called TDR for Time Domain Reflectometry.
Thanks for the addition. Just follow these steps:
simulate this circuit – Schematic created using CircuitLab
At this point, you've only one unknown node voltage, \$V_x\$, which is now easily computed using nodal analysis (or by simple inspection.) With \$V_x\$ in hand, you can work out all the currents in resistors \$R_1\$ through \$R_4\$.