The third gas man came today.
The second one left us spare fuses. One went yesterday, we replaced it, and then this morning it was off again. Replaced the fuse and half an hour later - it was out.
It is a process of elimination, so he didn't repeat what the previous 2 had done. The problem lies in the pipes being clogged up with something called magnetite (sounds like something from Superman to me) but the magnet stuck to a copper pipe so clear evidence of metal inside there. This causes the system to get clogged up, get too much and too little air, etc etc. The bad news is the cost to fix it.
It has to be flushed out which apparently takes a full day, two weeks after the chemicals are put in. The guy is going to replace the electric thing anyway, but I have decided to go ahead and get the job done by British Gas, because although they will cost more (the engineer said that) they then give you a lifetime guarantee on it. £700.
The laptop went into the repair shop yesterday. The guy in the shop said I had done all the things that he would have done to eliminate possible causes of the machine not connecting to the internet. Needless to say it worked just fine for him. Having not worked all last week - at home and at a wifi spot - Friday night it worked! The threat of going away was clearly sufficient.
However I have sent it away for a diagnostic test. The most obvious cause is a poor connection to the airport chip. Hopefully that isn't expensive to fix. On the plus side there were little chips on the keyboard which is a known fault, so they are going to replace that anyway.