EEA Partner visa expiring soon, what next?

I'm an Irish national living in the UK for 5 years with my non-EEA wife. She currently has an EEA partner visa that is going to expire soon and we're wondering which route to continue with (preferably the easiest and cheapest!)

Is it possible to get an extension on her existing EEA partner visa or apply for another one from within the UK?

If not, does she need to apply for permanent residence to continue living here and if so which form is the correct one? I have seen a few different ones and I'm not sure which one is correct (EEA (PR), EEA (FM))

She's also hasn't been working (looking after our UK born kids, however they don't have UK passports) - is that going to be a factor on whether she can apply for permanent residency? Would she be required to take a language test?

(Sorry if this has been posted numerous times, I tried a quick search but couldn't find anything specific)

If you have been living in the UK for five years as a qualified person with less than six months of absent with year with your wife she should have acquired PR and can apply for that. Because your children were born in the UK, As an Irish citizen, you are already deemed to hold settled status in the UK via the common travel agreement between the UK and Ireland and they would be British Citizens otherwise if you were born in Ireland they will be Irish by descent and can apply for Irish passports and hold settled status in the UK that way. Hope this helps a little.

Also, there is no EEA partner visa, an EEA family permit or residence card is issued under the EEA Treaty rights and does not count as a UK Visa.

Were the kids born in the UK at a time when the OP resided in the UK, and held Irish Citizenship?

HAs the OP been exercising treaty rights?

Sorry! - yes my wife is on an EEA family permit.

Yes, our kids were born in the UK when we resided here, they have Irish passports simply because at the time it was easier for us to apply for them.

I think I've been exercising my treaty rights, insofar as I've been working here for the last five years - is that what it amounts to?

In regards to applying for PR, am I right in thinking that I myself could apply online on the offical website and add my wife's details as part of the application, rather than my wife having to go through that (rather intimidating) 100 page EEA (FM) document (if that's the correct one)? (https://www.gov.uk/apply-for-a-uk-residence-card/apply)

Thanks!