I want to invite my former nanny (Brazilian) to the UK to stay with me and my family for a period of time. She will come as my guest and stay with me and I will support her. She only has about £2,000 in savings which is what she received from me when I had to dismiss her when I left the country. Will immigration believe she is here as a guest and not going to work for me. Would she be better applying for a visa for a short amount of time?
You want your former nanny to come to the UK using the Standard Visitor Visa as the vehicle for getting her in. You will sponsor the application. This is a very bad idea.
There were lots of people doing this: domestic workers were brought in by their sponsor, their passports were confiscated, they lived in slave-like conditions and overstayed their visitor visas because they were unable to leave their employer's household. A lot of women ended up in hospitals or safe houses for domestic violence. The violence and exploitation that these people were subjected to created a public outcry.
Parliament finally got involved and passed the Modern Slavery Act 2015.
The result is that the importing of domestic workers via the visitor programme is one of the UK's "hot buttons". They are on the alert for it and once the ECO (or IO) sees what's going on, they will refuse the application full stop. And since it's a 'hot button' they may send an inspector around to see if you have any other undocumented workers in your household. And finally, your credibility as a sponsor will be shot to hell and hence it's a bad idea.
Your nanny is Brazilian and can get in without a visa. In that case when you are finally caught they will throw in a harbouring charge in addition to the standard GBP 10k per person fine for employing an illegal.
If you want to do things legally, your nanny can use the Overseas Domestic Worker Programme. This is a closely managed inward route that nannies can benefit from (if you and she can jump through all the hurdles). I know it's a pain, but otherwise you are doing things illegally.
For your specific questions...
How long can you stay in the UK if you are supported by non-family UK citizen?
Following the rules laid down in Appendix V, a visitor can stay up to 6 months.
Will immigration believe she is here as a guest and not going to work for me.
In all likelihood, no they will not. It is not the historical pattern of domestic servants entering the UK as visitors to actually engage as tourists. Instead they are put to work by their employers and overstay their visas and this is accompanied by violence and exploitation. While in some cases the intent to visit may be genuine, the authorities have to observe the known pattern.
Would she be better applying for a visa for a short amount of time?
No. Even if she applies for a visa of one day, or even one minute, you cannot escape the pattern that you are attempting to import a domestic for the purposes of abuse and exploitation. As mentioned above, it is a "hot button".
Having said all of that, this site's remit is about travel. You can ask further questions about bringing your nanny to the UK in our sister site, Expats. They are a great resource for questions such as yours.