A picture really would help, but I think I understand what you have done, and what your questions are.

Can a large cutting of Euphorbia re-root? Almost certainly yes. If there is no rot, a large section of Euphorbia can easily stay alive for a year or more with no roots, and subsequently no water.

The actions you took, starting with doing some research of your own, were all correct, and will give your plant the best possible chance for survival. Rooting hormone can help, but only when applied to a fresh cut. It sounds like yours has already callused over, and I wouldn't bother cutting it again just to apply rooting hormone.

After you cut off/out the rotting flesh, you have to let the wounds dry and callus over. This can easily take weeks, especially in a cold, damp climate like I imagine you live in. Just keep it cool, but not cold, and also keep it dry.

Once all the wounds are callused over (the rule of thumb is one week per inch of diamter), you can pot the plant into a soil mix. Euphorbias need a very coarse, free draining mix. For rooting, I would use pure pumice (not avaiable on your side of the pond, but for us yanks its the best) perlite, turface, or similar. Don't bother with any organic components. As you already mentioned, the plant is top heavy and without a base. Make sure to support or brace it some way to keep it from falling over. I use heavy gravel in the pot to act as a ballast or counterweight, and for really big plants, some sort of guy wires. Its ugly and unwieldy, but it works.

The pot you use shouldn't be much bigger in diameter than the stem itself and shouldn't be too deep either. If the stem was 4" for example, a 6" standard or a 6" rose pot half filled with gravel would be my choice. I promise you it will look really silly, but once it has regrown some roots, you can transplant into a more appropriate looking pot.

After you've potted the cutting into your coarse, free draining mix, you'll just have to be really, really patient. And whatever you do, don't give it a single drop of water until you are *certain* there is new growth at the growing points. If you water too early, it will start the rot all over again. Place it in a warm (but not hot), and bright location. Direct sun is probably a bit strong, but then again, that far North, it probably wouldn't hurt.

Patience really is the key here. It probably won't do a thing until your weather warms up, and the daylight increases. Even then it may take months, if not a *year* to grow roots again. Fight the temptation to water! After a couple months, you can very carefully offer it sips of water, but don't soak the soil until you are absolutely certain there are roots, as evidenced by the new growth up top.

In the mean time, be on the look out for more rot. If you missed a rotting section, or if the problem is more systemic, you may find more surgery is required. As a last ditch effort, you can cut the last few inches of each growing point and root them in the same fashion. At this point though, that is probably too drastic. Again, keep an eye out for rot progressing up from where the roots used to be.
Now, be warned that all of this advice will be useless if your plant was already in very poor health. Unfortunately, these type of plants don't do very well grown indoors, and will usually end up weaker and more disease prone because of it. Then again, if it has gone 10 years, it couldn't have been doing that bad.

If you have any other questions, or if something wasn't clear, please ask. It really is a shame to lose such a close "friend," and none of us want to see that!