After probably a month of not uploading videos to Youtube and getting desperately behind on my playthroughs, I finally submitted and went to my local nerd store* and talked to the nerdling-in-chief about my problems.
While a laptop upgrade is out of the question, for laptop design reasons, they are, however, selling their in-store computers which they use for their "LAN parties" and "48 hour Battlefield 4 marathon sessions" at a suitable discount price. Top-end market graphics card included.
Obviously, this doesn't solve my internet issue, and will require some setting up and getting used to (not least re-downloading all my games from Steam and Origin). However, it should be able to run any currently released game at 60 FPS on high graphics settings, and will probably turn things like Dragon Age: Origins and Skyrim into virtual works of art.
So, some good news, at last.
*it really is a nerd store. Combined computer gaming and tabletop RPG and tabletop gaming needs, all under one roof. They have a homebrew Warhammer/Dungeons and Dragons campaign setting, for Christ's sake. Also, neckbeards everywhere