How much higher would the VAT have to be to add in universal health care?

There are probably two issues to address in the US.

A lot of people are uninsured or have to pay out of pocket themselves (for insurance, treatment or both), leading to all kinds of unpleasant consequences like random bankruptcies and deaths from lack of basic healthcare, or economic effects like people who should change jobs or start new companies but can't afford to. Changing who pays is something that could probably be solved by a single payer model as in many other countries. Funding it with a national sales tax would probably be doable. But it might not help that much.

The other bigger issue is that the US healthcare sector is spectacularly badly structured. The US government spends more per capita on public healthcare than Canada does, then has a huge private system on top. The total spending is staggering, largely because the incentives and structures are skewed. Sadly, there are probably way, way too many vested interests to allow the possibility of ever changing that. There are probably some things that would help, but anything you touch is ruining the livelihood of someone, and probably someone with an effective lobbyist. (note, this is a problem that several countries have run into: once a health care system is up and running, you tend to be stuck with it). I'd probably start with something small, like forcing public disclosure of the price paid for every drug, by customer, at every stage of distribution and payment.

For fun really radical ideas, I read a paper that argued out that drug patents might not be a good idea in the US. The obvious retort is that without the prospect of profits, the drug companies would have no incentive to do the expensive research, which seems like pretty compelling logic on the face of it. But the weird thing is that their aggregate drug research budgets are way smaller than their aggregate extra profits from patent exclusivity; for society as a whole, it would be much cheaper for the government to pay for all the R&D and have no drug patents at all. This would also remove the bias towards research on diseases of rich people who can pay a lot for a drug relative to research targeted solely based on the biggest aggregate disease burden. It would naturally go along with publishing all research results, positive and negative.

Jim