The deprivation of capital rules don't require you to prove what you did with the money. She can challenge the decision on the basis that it all went on living expenses and as a pp has said, £30k over 3 years isn't excessive at all.
The rules also only apply if there is a perception that the money has been intentionally disposed of with the intention of maximising benefit income.
If I were advising her, I would go through the bank statements of both accounts with a fine toothed comb, and account for as much as I possibly could. It is perfectly reasonable to have a joint account with someone with whom you share a home, so that shouldn't be an obstacle.
The joint account should show what has been paid out in rent, deposits and agent's fees, what has been spent on council tax, utilities etc, shopping and any furniture or household appliances bought jointly. That'll come to a goodly chunk. Then there's any car expenses, clothes, holidays - all of these are an acceptable use of capital if they're reasonable and not excessively extravagant. Buying a Bentley would be seen as excessive, a Ford Focus wouldn't. Spending £50 having your hair done every few weeks would be fine, spending hundreds getting Nicky Clarke to do it might be excessive. In one case I dealt with, the council accepted £3k or thereabouts spent on getting the client's teeth straightened as reasonable.
Gifts can also be ok, again, providing they're not extravagant. In another case of mine, the council accepted that the client gave each of her 4 adult children £5k each from her late husband's death in service benefit, because they had always agreed that if anything happened to either of them, they would make sure the kids were ok.
Other things I've had accepted as reasonable was some expensive, professional quality sewing stuff (overlocker, sewing machine etc) purchased by a client with a view to making some money from dressmaking, garden tools, a camera that cost several hundred and a trip to Australia to visit family.
All my deprivation cases pre-date universal credit (we haven't gone full serivce for UC yet in my area), and I always started off dealing with the council, as they tend to be more reasonable and treat people as individuals. You can actually go and meet them, armed with a schedule of spending, bank statements and receipts. Once I'd established the case with the council for HB, the DWP just rolled over on everything else.
I'd also play the "naive" card, and say that the adult child took advantage for any significant funds that can't be accounted for.
Most important is to challenge the decision within the one month deadline, you can do that with a letter and provide full evidence a couple of weeks later.
Make it as easy as possible for the person reconsidering the decision, with a nice, neat schedule of where the money's gone.