I'm sorry everything is hitting you all at once and making life impossible, and making it seem like any dreams you have are impossible. I've been there, it hurts. Whatever you decide to do ultimately I hope it's what you feel is best for you, best wishes whatever route you go.
I assume this assignment is an application for university and you're not already going there? If you were in university they usually will have laptops you can borrow. If not try libraries and they might have desktop computers or laptops. You could also reach out to the university and explain your situation and see if they'd make an exception and allow you to use their computers as a guest or something. I've never been to England to know but do you have any internet cafes or places like that? Where you could pay to use a computer if there isn't a library or other place you can borrow one from.
For your broken laptop did you have information on it that is essential? Does it still work other than the screen? If there is essential information on it and it doesn't work anymore you can take out the hard drive and they have cords made so you can usb it to another computer and access all the info on the hard drive. This might take too long though - I had to order one online when my laptop broke a year ago, and also had a friend with tech knowledge tell me which would work for my model of laptop. If it still works other than the screen as others suggested you could hook it up to an external monitor. I don't know your laptop type but even without a video port it *might* be possible. Usually it's HDMI to HDMI to connect screens but they make all sorts of cords for that too so it might be possible to get HDMI to ??? where ??? connects to another suitable port your computer offers. Towards this end you may want to look up if your specific laptop can be wired to an external monitor and see what the internet has to say.
You could possibly continue work on the papers on your phone (I assume that's how you're writing here). You could write it in a note app or write it in an email to yourself, as long as you have email on your phone you can easily put your writing there so when you have access to a computer again you have your writing on email and can copy/paste into a proper document format. If you have a google account you can access their Google docs online, they have an app and you can also pull up google docs via the Chrome web browser (and potentially others) - editing sucks here though. When necessary you can go into the Chrome browser settings and set it to desktop mode if you need to trick your phone into thinking it's a computer, this can make some things a pain to use but just fyi if you didn't know and it could be helpful. Ahem I'll stop rambling now.
All just thoughts and I'm wishing you the best and hoping you can overcome this and see to your dreams. I've given up on most (all?) my dreams but I'd much rather others get a shot at theirs if they have them. I won't be online much longer today but if me writing too much can be helpful just let me know. Good luck and I'm sorry for everything that's happened and about your laptop.