I think this too...Looks to me like C13 is the designation of the capacitor next to C12. Dunno about the footprint, but looks like there would be a transceiver IC soldered in those 12 thruholes if 10bT was supported.
A 10bT LC NIC will likely cost less than a transceiver box and will definitely be more convenient.
Reworking the card and adding the resistors isn't a problem for me, but I'm doubtful about the ROM... I think I'll put away this card...Yeah, it seems that just soldering on an RJ45 wouldn't be enough. As jt says, the ROMs are most likely different, as the labels suggest. Likely, you could burn a new ROM, but that's probably not enough. One resistor bank is completely missing on your card, and I can spy different values on at least one resistor on the other bank. Likely, you'd have to rework those as well.
With as cheap as LC network cards are, best just to buy one with the port you want.