The question here seems to literally be "What CAN I do?"

OP does not have perfect vision.

Cannot walk very far.

Needs limited exposure to stressful situations.

Will not use public bathing facilities, nor traditional toilets.

I've been in the - cannot walk far - category and also looked for minimum exposure to stress and anxiety. When I visited Japan alone with these expectations I didn't go to Tokyo but spent time on an island near Okayama as well as visiting Nara.

If I had needed to go to Tokyo I would find somewhere to stay near Nippori and just potter around the Yanaka, Nezu and Sendagi neighbourhoods. Occasionally, whenever I felt brave or strong enough to do so, I would get the Yamanote line and visit another area, either to see the gardens or museums or whatever sights and attractions I was interested in.
If you need ideas or inspiration then check out what the Grutt pass will give you access to:
http://www.rekibun.or.jp/grutto/pdf/grutt_pass2014.pdf. It's 2,000 yen, gives you plenty to do for any of the 16 days when it is raining or cold and miserable and (with the exceptions requiring a supplement amount) you won't need to pay anything extra on top of the travel costs.

A big advantage of visiting a city rather than the countryside is that you are never far from a taxi if you get into problems.