Does a domestic worker from abroad need to pay NI and tax?

If someone is resident in the UK, then they are liable for UK tax. Residence for tax purposes is different from residence for immigration purposes. To be resident here for tax, you have to just be present in the UK.
The annual tax free amount is £6475 this year and after that tax is payable at 20%. The first £6035 of pay is free from NI, after that, it is charged at 11%. The employer is responsible to deducting tax and NIC from the employee's wages.

Yes, they do

if they are earning their income here,
then they pay taxes and NI just like everyone else

The amount that is deductable depends on their income

You first have to establish your residency (which isn't the same as residentship) position with form P86. Otherwise, as a non-British subject, you aren't entitled to any British tax-free allowances, and employer should deduct Basic Rate (a straight 20%). You can download from http://www.hmrc.gov.uk , then put "form P86" in search box. If that is OK, you get £6475 a year tax-free. That is proportioned throughout the year, and after that 20% tax is deducted. Tax is calculated on a cumulative basis. The UK tax year goes from 6th April to 5th April.
NIC isn't cumulative. Each payday is taken in isolation. Once your earnings reach £110 a week, 11% is deducted of the excess.

anyone who is employed in the uk whatever their nationality has to pay national insurance and tax (if their earnings are above a certain level), if they are working cash in hand and not paying it then it is illegal. it will be automatically deducted from their pay by their employer who also makes a contribution towards it.

everyone who works has to pay tax and national insurance....

how much depens on how much you earn as its worked out on a percentage of your wages. for instance my husband pays £80 a week in tax and national insurance

Yes.