If I have to flush out a tank, I do a series of 50% water changes, spaced out at least a few hours apart. Dechlorinate each time. After 5 or 6 over 3 days or so, you should be good. I wouldn't have advised salt for this issue (we have this debate often - put 3 aquarists in a room and you have 7 mostly grounded opinions) because it is hard to get out of the water. But you can do it.
For the future, my last test kit expired in 1992, 3 houses back. I haven't missed it. My not using one has nothing to to do with experience. By the time you pull the thing out, do all the tests etc, you could have done a water change. If you do 25 to 30% regularly, at least once a week, you avoid most problems. If you do the test and don't like the results, you do a water change anyway, right?
The tests are an unnecessary step if you maintain the tank. Some people have the misfortune of living in agricultural areas where their tapwater nitrate readings show it to be polluted and dangerous for fish, and for them, API type kits are needed as they have to deal with the bad water. Somehow, their extreme situation has been projected as normal by test kit marketing. The last time I was in London it didn't look like North Dakota...
London water is hard, and it might lead to Bettas living 3 or 4 years instead of five. But most store bettas have shortened lives from how they are raised anyway. It sounds cold to say it, but it's reality. If your fish went from fin rot to a bacterial infection, then I'll hazard a guess you bought it with problems brewing.
I'm sorry this one died. You have a sharp eye for problems and gave her a good home.