Thank you for asking your query.
I am sorry to hear about your in-laws medical condition and will try to assist you my best.
From your description she seems to be more or less in a persistent vegetative state. A vegetatvie state of this long duration will have definitely to do with a damage in the brain. There are many problems which can cause coma. Ranging from metabolic, hormonal problems to neurological problems. But all other than neurological issues will either progress or get better in some duration. If there is neurological problem the comma can persist.
So there could be a stroke or a brain infection like encephalitis which led to the coma in the first place. I am afraid by this time a lot irreversible damage might be done and recovery might not be possible.
Now coming to your specific query , what kind of tests can be done :
1. MRI of the brain
2. Blood tests, Electrolytes, Renal and Liver function tests, Thyroid function tests.
These basic tests will give a relatively fair idea on how things are with her now. If needed further tests can be done after analyzing this data.
Hope this helps.
Please feel free to ask follow-up questions if any via this email.