It's been a while since I flew an Angelflight, so maybe things have changed. But Angelflight has been mentioned several times, though this type of trip really isn't either the ideal setup for Angelflight, nor really the intended use.
First, this is a LONG flight - 1400 nm or so. Although some Angelflight pilots do fly jets, the typical Angelflight mission is performed in light single and twin-engine pistons. And with a flight of this length, it would be multiple flights strung together. Often in non-pressurized, noisy aircraft. Literally, this could be an all-day, miserable trip for someone with post-stroke symptoms.
Second, the intent of Angelflight is for getting patients to where they receive medical care on a non-emergency basis. This doesn't sound like that at all, this is simple A to B transportation. Unless things have changed with AF, I'd guess that this wouldn't meet the criteria for an AF at all.
The OP asked about private jets that just happen to be going that way. Sure, that's possible, although it will take some digging to find, since of course such flights aren't scheduled on a website.
There's always the charter route. No mention was made of the couple's finances, but I'd estimate somewhere in the $40k or so range. This would definitely be the most comfortable and convenient option if they can afford it.
Sometimes charter operators have deals on "empty legs", where they need to reposition an aircraft, you'd have to contact them about that.
Or, you know, the airlines, as much as we like to hate them here, are actually pretty good at what they do. And they often are the best solution. Looks like there are multiple options for direct flights from DEN to EWR for just $200 a person in the next couple of weeks, and flight times of 3.5-ish hours. That's pretty good.
