Is the job you're quitting a full time job with benefits? Would you lose any vacation pay accrued?

If not, I say just talk to the manager and have your next day be your last day. Yeah it's bad to not complete your 2 weeks, but if this is causing you stress AND depriving you of sleep it's not good. It will only hurt your new job. It would have been nice for them to allow you 2 weeks instead of starting immediately.