From what I've read about getting committed is that you're out after a few days, though that could be they had been suicide attempts and they wanted out, and under EU law I don't believe they can be kept against their will for more more than three days unless they are deemed a threat to others, though don't quote me on that. I saw that you're American and process may be different over there.
Just read you could be there for seven months, which is lucky. Just remember you'll need to come out sooner or later and moving back into your mum's house could prove tricky. Having no money when you want to die does further complicate the situation as the number of options you have are limited, and you won't even have money to buy some pain killers.
I would either go to the clinic, or alternatively just keep trying the job-hunting, any job will do. Both would possibly introduce you to decent people who could help you, the former would be trained professions who may even be able to help you find work or accommodation, the latter would give you money and you'll feel at least you're of some worth and there's always good people around. You'll recognise the good people as generally they are quite quiet and reserved, and you may need to initiate small talk. There's no place I've ever worked in that has been made up solely of idiots, there are always at least one or two kind people.
If you opt for the clinic, which from the outset seems the better option, then continue to job hunt and make it absolutely clear to the nurse that you are desperate to get on your feet and working. She may be able to find you work while you're staying at the clinic.
Being broke and unemployed with a mum who wants you out are external situations that I fully appreciate must be difficult, but remember they are external. And I do believe many external situations can be fixed, especially when you're young, though it's not guaranteed. Even though you wouldn't be able to take your life in the clinic it does give you time to get treatment, and with any luck, a job and finances. Then you can take things from there...