Sell the existing PC (including the 110GB SSD and GTX 960 but keep the 1TB HDD for your new system) to help finance a nicer video card.
That way you'll end up with a single system that is much better for both playing games and streaming, than keeping two systems (and having to buy a cheap graphics card for your old system).
With the money left over from your £900 budget and the proceeds from the sale of existing system you could be in GTX 1070 purchase territory.
If you go this route, you'll have to find out two more specs of the present system in order to be able to advertise and sell it - motherboard and PSU. For motherboard, just download CPU-Z and it will tell you. For PSU, open the side panel of the case and have a look.
Alternatively, if you log into your OverclockersUK shop account, you should be able to find the specs, or the name of the system, in the "My Orders" section of your profile.