Why pay PC World?
It's easy enough.
Download malwarebytes and run a scan.
And/or superantispyware and its scan.
They will check nothing slipped by AVG.
(is it up to date?)
You could then use another antivirus for real belt and braces.
eg Avira - AntiVir Personal - Free
All available from click here
Save downloads to a flash card as well for future easy use.
In view of your flaky PCs and Mum wants a new one, get one. Unless it breaks the bank.
Personally I would go for a sub-£400 laptop with W7 64 bit, 3GB RAM, 250GB HDD, as a min.
(check printer and modem support W7)
I assume your Mum is not a games monster.
Don't pay for Norton or McAffee on it.
Some suppliers will knock £30 off or so without it.
AVG or Avira are fine.
Assuming you get a new computer:
Make sure Antivirus is installed as soon as you get it running (Use AVG or Avira that you saved on the flash card)
Note Windows often says "Not recognised or installed" to a card when you plug it in.
Try again, a couple of times or another USB port.
Also for W7, turn UAC onto max.
Start Menu, type UAC in the search box and click the icon that pops up.
If it annoys you, you can reduce it one notch when you are familiar with it.
If you are not transferring too much data, you could use flash cards.
Or buy an Ethernet Crossover cable and link the two computers directly.
You would need to run Network Setup Wizard on an old XP computer.
New W7 may sort itself out automatically.
Google "peer to peer" network if a problem.