Replacing old home telephone wire?

So I tested the little box marked 'phone', 'line' and 'modem' and it works. However, connecting my modem to it would not be feasible since I'd have to run long ethernet cables through the laundry/boiler room doorway, and down a hallway and into the living room where my desk and PC are.

I talked with my ISP again, and told them I had narrowed down the problem to an issue with the telephone line behind this wall jack (some kind of combination of coax cable and phone cable jack)  in the dining room:

The ISP said they could send out a tech to fix it, but it would cost $125./hr. I don't think the tech would be able to replace the telephone line anyway, as I previously mentioned, if the line is stapled to a stud, they wouldn't be able to pull it out without ripping out the dry wall.

This is the wall jack in the kitchen:

I managed to run the CAT5e cable from the kitchen wall outlet all the way around the living room to my desk (on the left) where my modem and PC are. But as you can see it means running it above the radiator. I think it will be okay though.

Since I am getting 44-45 Mbps (not ideal, but good enough), I've decided to live with it this way.

Thanks to everyone on this thread who helped me with this problem!  I may not have bothered to try and test the kitchen wall jack otherwise.