The first thing she and you need to determine is if the TV has an Analog (meaning not digital) AUDIO OUTPUT. This will take the form of a pair of Red and White RCA-style Phono-type connecting, or it will be found in the SCART connector and can be accessed with a simple £5 adapter. The final alternative is to tap into the Headphone output.
Next is a remote volume control. The Audio Out on a TV are usually
fixed level , meaning that output does not change with changes to the TV volume control. You must use the volume and remote that come with your audio amp.
In some cases, the Headphone output of the TV is connected directly to the TV volume, in other cases the Headphone volume is set in the menus somewhere. Still the Headphone out can be a good workable option if nothing else is available.
Note while external active speakers are a workable option, very few of the come with a remote control.
From you statement, we assume you need just an amp with multiple inputs, and a compact pair of speakers, and with a total price of about £250.
Here is one amp to consider -
Pioneer A209E, 35w/ch, £139-
PIONEER A209R AMPLIFIER - available from Superfi UK
If you can push a little higher, the next model up is also worth considering -
Pioneer A307R, 45w/ch, £179-
PIONEER A307R AMPLIFIER - available from Superfi UK
For speakers, while there are many, these are something of a bargain in a good speaker -
Diamond 9.0, 4" woofer, 60hz, £59/pr-
Wharfedale DIAMOND 9.0 Black | Speakers Per Pair | Richer Sounds
A larger alternative and still close to your budget is the -
Diamond 9.1, 5" woofer, 50hz, £99/pr -
WHARFEDALE DIAMOND 9.1 SPEAKERS (PAIR) - available from Superfi UK
Both these speakers are on close-out and therefore deeply discounted, but that also means limited stock and limit colors, but none the less both are speakers worthy of consideration.
As an alternative, you might be able to find the Monitor Audio BR1 for about £100 per pair. Again, on close-out and deeply discounted. For a larger speakers with better bass response, consider the Monitor Audio BR2 for roughly £150/pr. Both these speakers are on close out, and again, that means limited stock and limited colors. (You should find both in a Google search, but PeterTyson.co.uk, QuantumElectronics.co.uk, and HiFix.co.uk are known suppliers.)
Another speaker, a current model at standard price, would be the Tannoy Mercury F1, which are about £100/pr.
This by no means touches on every possibility, but it should help you get started.
All products listed with the exception of the Wharfedale Diamonds, are available from multiple sources, and should be very easy to find.
Steve/bluewizard