£300 is the budget, so 8GB is probably not realistic unless sacrifices are made in other areas. An SSD will make the performance snappy but good graphics performance is necessary to stream videos. My little ASUS Transformer does everything I need EXCEPT it won't stream video to the Chromecast on my TV.
eBay has a very good set of filters where you can specify screen size, memory and storage to see where to spend your money.
You might consider a computer shop for a recent, but used, machine. I'd suggest a 13" screen but compare that to what you have - it is the diagonal measurement of the screen. I'd recommend at least 100GB of storage.
Does the school not have recommendations or access to educational discounts?