How long does the investigation by department of aging take?

You are doing all you can. Be the squeaky wheel. There is little else you can do. They may be checking reported accounts, and etc. I think there is a lot you likely don't know. Consider contacting your Congressman or woman. Wishing you good luck.

Investigations like these can take years and usually nothing comes of them. The chances of there being any money left are slim to none unless you're millionaires and a court-order has frozen accounts pending investigation.
The investigation I have going on with the state over elder financial abuse concerning the nursing home my father was in for a few months is still supposedly "going on". He's been dead four years. The investigation over his then girlfriend running up his credit cards was never concluded. She received zero consequences and got away with all of it. The only time I ever got the slightest update is if I'm able to put in a whole day of being on hold, getting transferred, listening to automated menus, and leaving voice mails. Once in a blue moon someone will leave a voice mail for me saying that it's still being investigated. I did this every Friday for two years because that was my day off. I leave a message on somebody's voicemail now maybe once a month just to let the investigator's office know that I haven't forgotten about it.
I know that if I swindled an entity or person for the amount the nursing home did, I'd be in jail. So would anyone else.
Your best bet would be to call the investigator/caseworker once in a while to let them know you're still here. Then forget about it, because chances are you won't get recover anything and no one will be brought to justice.