First of all, keep in mind that I am unable to diagnose you because I am unable to examine you, this forum is for educational purposes.
The symptoms that you describe are non-specific for any single disease, but I would suggest that you see a neurologist soon to repeat your workup. Buzzing in the ears and dizziness is common with ear (middle/inner) infections (viral/bacerial, etc) and can also trigger headaches and nausea (also called viral labrynthitis). However, this condition would not cause hand weakness and this is concerning for more serious problems (stroke, tumor, brain abscess, etc.) If you have had the left hand weakness for a while, it may be due to a pinched nerve in your neck or carpal tunnel syndrome as well. I would suggest that you get a hearing test (viral labrynthitis will often cuased decreased hearing), elctro-nystagnogram (to evaluate for the cause of dizziness), MRI of the brain and cervical spine with contrast (to look for a structural lesion such as stroke, tumor, disc bulge etc), and an EMG of the Left upper extremity( to look for pinched nerve in the neck and carpal tunnel syndrome).
I hope this has been helpful.
Hi. Please see comment I posted today to lilpanda. Could pertain to you.
Lynda