Visa-wise, your situation is a mess because (thanks J...)
There is no provision in the Immigration Rules for leave to enter to be granted solely to allow a person to care for a friend or relative in the UK. *(document linked later)
Interpretation: they wish to allow it, but immigration law has no idea how to treat it.
You have a long term health issue so I presume you'll need long-term support, which means a Visitor visa is dicey. You can't chain visits, because UK will stop "living in the UK for long periods of time through frequent visits". Using it is a short-term band-aid that will paint you into a corner later.
Further, what you're looking for is an aide, nanny, or assistant. That is employment. Your entire venture will fail if you do not understand that fact. Now, there's a long-term visa for family members, and once thus landed, obviously a family member can provide that support. However the application is expensive and slow.
As a short-term visitor
Border Force will have trouble believing a Southeast Asian of limited means wants to vacation in the UK for months. So the temptation is to claim a short e.g. 7-day vacation. But that creates a piccadillo: do you leave in 7 days to avoid a soft overstay (of your stated intents)? You can get away with that for 3 months, but then, you are unlikely to be approved for a future entry unless the soft overstay can be explained well.
On your second entry, how do you explain to the immigration officer how you blew right through your stated plans, and stayed right up to the hard overstay limit? This will destroy your credibility for a future entry. Do you stay the full 3 months of the visa and then leave (to avoid a hard overstay), in which case you cannot explain the substantial soft overstay? Or do you simply abscond on the visa, and stay indefinitely, and duck into an alley every time you see a Border Force truck? None of these are good options.
A misstep here - abusing a visitor visa - will disqualify your relative for the other options, and place in suspicion anyone else you try to bring in. So choose wisely.
Where the Entry Clearance officer feels that an applicant seeking entry to care for a relative does not qualify as a visitor, but there are compelling and compassionate circumstances, the case can be referred to, Referred Cases Unit, UK Border Agency International Group to be considered outside the Immigration Rules.
This is essentially "throwing yourself at the mercy of the immigration system", but that is sensible, because this type of case does not fit anything well: it's clearly not short term leisure/vacation, but it is also not quite what the employment rules intend. And as said at the start, the Immigration Rules don't know what to do with family caregiving.
As a family member
Look into Part 8 (family members). The difficulty with this type of visa is the high cost and long wait for approval. From Border Force's perspective, their mission is to create some path of resistance to prevent what the more xenophobic Yankees call "chain migration", where one person brings in their family, who bring in their family, who bring in their family etc. So this process is made expensive, slow and uncomfortable to provide a general deterrence.
You would have a painful 1000 quid application fee and a wait of several months.
Unfortunately, a person can't stay in the UK on a visitor visa while this is sorted out; they're expected to stay in their home country during the application process (and may be called into the embassy/consulate there).
So if you have a second family member who could caretake while your mother-in-law goes through this process, that may help.
It is worth it to pursue, however. Once it is obtained, then the question of a long-duration stay, and of doing domestic chores within the family, is settled.
As a worker
The problem with a visitor visa is that as far as they're concerned, sneaking in family to work in a family business is the oldest trick in the book. It is one they are familiar with and on the lookout for. Especially when it's a big rush, because that typically means a restaurant lost a cook or dishwasher and needs a replacement ASAP. What inevitably follows is the worker staying indefinitely, being paid under the table, employment taxes not paid, and healthcare coming off the dole.
Look intoPart 5 (domestic workers in private households), note that "raising a daughter" easily counts as work history under the circumstances. Note also 159G in part 5.
"But I aim to pay them nothing" - that makes no difference in employment law. It actually makes it worse as far as Border Force is concerned, because part of their remit is to combat human trafficking, and so they need to diligently research every such claim to exclude that.
The correct way to do this is either
apply for the correct working visa for the family member. This visa will have a much longer duration, so you won't have the above problem, and be renewable potentially indefinitely.
as they intend, hire a British citizen for this task; or an EU citizen up until Brexit. As for the cost, they expect you will avail yourself of any and all government assistance programs offered for this.
I agree with Michael's answer that there's no problem applying for a visa much sooner than 3 months. Even going with the normal no-rush option, my wife's nephew got a month long visa sorted in a few weeks this summer.
I'd agree with the Harper's answer that if your intent is for your mother in law to do any kind of work, I'd not apply for a visitors visa but rather look for other alternatives.
But, if your wife would be happy to just have some company in a stressful time, I'd just keep it short and clear by saying that you want to bring your mother in law over to visit the family and maybe do a little light sightseeing or similar.
However, I have no idea why Harper talks about a 7 day visit, if anything I'd say that such a short visit would be very unusual for a family member visiting and might make the border force suspicious. At least from Thailand, there's nothing unusual for family members to come and visit for multiple months, I've had my mother in law (also Thai and with with very limited means of her own), come over and visit for month long visits twice without any issues at all. Many other friends and relatives have brought over family members for extended periods of time and I've never heard anyone mention any issues with the border force.
As long as you can show that you have the means to support her properly and that she'll have suitable travel insurance etc, and, whatever you do, don't lie or exaggerate any details on the application (income or similar), they do check up on stuff and any lying will probably mean an instant denial.