Something like this should be a simple application, but it's not. The UKVI have made it complicated. Firstly, you can, of course, apply for an extension of the visit visa, but it may or may not be granted. The "compassionate" UKVI take the view that a spouse or family member cannot remain in the UK to care for a sick realtive or spouse, but might be allowed to remain in order to arrange long-term care. Compassionate, indeed !
As theoldgit says above, the best route is probably to apply under the family life route. There is an element of "the right to family life" in such a human rights application. Basically, the argument is along the lines that, as a married couple, you have the right to family life together. And, as you cannot be expected to leave the UK, your wife should be allowed to remain with you in the UK. You would still be expected to meet the financial requirements of the immigration rules, but if you cannot do so, your wife would probably be put on a 10-year route to indefinite leave to remain, rather than the normal 5-year route.
I would advise you to take professional advice, from a UK immigration advisor (expensive) or from the Citizens Advice Bureau (free). All advisors in the UK are either registered with the OISC or are lawyers. Non - professional advisors are illegal. There is one (maybe more) OISC registered immigration advisor in Thailand, based in Pattaya, and I can pass his details to you if you wish.