The Mrs isn't working and is eligible for any benefits, so no point in signing on, although as I understand it, they would pay her NI.
She only needs about 6 more years of qualifying years to qualify for the state pension.
Ideally she would like to avoid the grief and hassle that signing on brings.
Now it appears that the rate would be £14 per week for a Class 3 contribution.
However, if she was self employed, this drops to just over £3 per week.
What's to stop her, declaring herself self employed and paying the lower contribution? Just submit a tax return each year, showing a modest part time business, that earns less than her tax allowance?
Before anyone calls me a tax cheat, the net cost to the state doing it this way, rather than signing on, will be reduced.
Has she recently left employment ?
Check to make sure if she's entitled to contribution based JSA, based upon your previous (recent) NI contributions.
https://www.gov.uk/jobseekers-allowance/what-youll-get
I paid a top up as I was out of the country for 2 years, it was quite simple with payment in a single cheque to cover the period and paid upon my return
I think doing as you say in creating your own "sole trader" business seems like quite a good idea.