Just option George: Book into a Travelodge in Luton then train into London for the galleries & museums etc. Cheaper and doable day trip.

Plenty of good suggestions and choice, how many months have you got 
There are some decisions to make, for all or segments, between the best cycling and visiting the most interesting places, my touring leans towards the former, for the latter I mix folding bike and the train or coach.  Andyb1's Easterly route has a lot going for it, plenty of interest, but not as scenic (Or hilly) as up through the Pennines. If visiting cathedrals is your thing, this is worth a look:
https://www.cyclinguk.org/cathedrals-cycle-route-challenge
As with the NCN, those routes are best used as a basis, rather than strictly adhered to.
The Scottish borders, as in South of the Glasgow/Edinburgh line, is an area I know little about, but having done a couple of Audax that passed through, they're on my list and there's centuries of history due to the English/Scottish animosity. The area is scenic in a rolling countryside way rather than the wilderness of the Highlands. IMO the North West of Scotland has the most scenic cycling in the UK, even if you don't jump across to any of the islands, but the things to do off the bike are few and far between.  Other parts of Scotland have some equally good bits, there's just some less good bits between them.
Couple of other random thoughts:
if you decide to ride out of London, the NCN route along the Lee Valley, from the Olympic Park in Stratford (Not Avon), is easy riding, mostly traffic free, and gets you well clear of the metropolis. 
I use trains quite a lot, mostly with a bike, it's a mixed bag, some considerably better than others.  I've never failed to get where I'm going, but it isn't always stress free!  If taking a London train from Edinburgh be sure to book one that originates there, that's most but not all.  it just gives you more time to load and you know the bike space won't already be full.  Another vote for booking with trainline, it's one of the few online options where you can reserve a bike space.
I'd second B cereus's point about trying to cram too much in, it's a trap I sometimes fall into, but exploring somewhere rather than passing through on a strict timetable usually, for me, results in more memorable tours.
If the Audax route you originally posted is more your thing, then have a search on RWGPS, for the LEL (London-Edinburgh-London) route.

‘Inconveniently located far from anything worth seeing.’

Have a look at how close it is to a London Underground station.   The tube system will get you to the areas you want to visit.

Thanks Danneaux for your helpful response.

I've tried creating a nipple of super glue around the end of some spare cable, and it does seem to work. Now, I just have to try this on the actual cable after it has been threaded through the splitter (I'm using the J-TEK splitter, which Thorn fitted to the bike).

The cable splitter lies approximately mid way between the cable stops when the Rohloff is in gear 7, so hopefully cable flex will be at a minimum. My tandem has the Ex box, so both cables run under the down tube to under the stokers seat tube, and onwards to the Ex box along the chain stay, which I believe is the optimal set up.

Once again thanks for your reply.

BR

Hugo, thinking a bit more about your problem, I'm wondering about the role of excess or repeated cable flex at or near the grub screw causing metal fatigue.

Of course, the rear splitter allows for actual un/screwing, as it is basically just a seat for the leaded end of the cable and this tolerates some "slop". The front splitter half is secured rigidly to the cable so any movement will be concentrated at the screw.

Besides reinforcing the stranded wire as I suggested earlier, is there something causing the front run of cable to bend and flex unduly? Something like a nearby cable stop or frame boss? Maybe a clamped-on accessory band? Is the anti-scratch o-ring still intact on the front half of the splitter? Is the cable end getting bent overmuch while the bike is disassembled or put into a case?

Just trying to think of possible contributors.

If your splitters have hex flats, it might be worth holding the front half stationary from twisting with a small wrench while you spin the freely turning rear half. I modified a small, automotive ignition points wrench for a friend's similar need and it seemed to do the trick for him.

EDIT 1: If you are using the J-TEK cable splitters without wrench flats, they're available in different lengths and for gear vs brake cables. I presume the difference is related to the size of the leaded cable end? Perhaps fitting a longer or shorter example -- if possible -- would ease stress on the cable. See...
https://www.sjscycles.co.uk/search/?term=j-tek%20cable%20splitter&geoc=US

EDIT 2: Is the cable splitter fitted midway in the open run of cable between stops? If the splitter is too close to the front stop, the cable will flex at greater angularity, right where it is clamped by the grub screw(s), resulting in metal fatigue and snapped cables.

Best, Dan.