I can help with this!
I have since moved out of the building, but I figured this out a little while ago and wish i knew about it a long time ago.
The difference between those two types (RJ-45 and RJ-11) are the number of copper wires that run inside it. RJ-11 is 6 wires, RJ-45 is 8.
In most modern buildings (20ish years maybe?) the telephone wires are usually made with the 8 connectors and only at the outlet do they convert to a 6 wire connector.
In my unit, there were cables for the phones that all connected in the furnace closet. I didn't realize these could have been used to deliver Internet by using the new connectors.
I don't know if your cables have all 8 pins however but I think they probably do.
This means that you can likely swap out the outlet. I did this exact thing in my current home and use it to get 1000 Mbps.
I used an AMPCOM toolless jack found here:
I also needed two more tools to get it done:
I don't live in Loganview anymore, but I have been meaning to run over there for an errand so I could stop by to loan you the tools this weekend if you want to borrow them.
John