When you say you can't get the nipple out... what are the rims ?
Is the inner wall of the rim taped ? Or are you saying the holes are 'blind' ?
Is it that you don't want the faff of taking the tyre off, the mess of sealant, taking the tape off to get to the internal spoke hole ? If so... unlucky! You'll probably have to !

If the internal rim wall is 'solid' with no drilling for accessing the nipples then how did the nipples get put in, in the 1st place ?
(Answer may be via the valve hole on the inside of the rim and shaken down to each of the nipple holes. I've seen that on a rim recently... total and utter PITA with no thought by the manufacturer as to maintenance.)

I've had a good few alloy nipples seize on spokes on mtb wheels and then need replacing when the spike broke in the thread. I always had to replace the nipple, I couldn't get the stubby little broken bit out of the nipple.
I always use brass nipples as replacements and when rebuilding a wheel - bollox to the 20g saving over alloy nipples, its just not worth it

It's usually fastest/easiest to just cut the spoke off as close to the rim as you can, and then replace both spoke and nipple.

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Tech writer/support on this here site. FIST school instructor and certified bike fitter. Formerly at Diamondback Bikes, LeMond Fitness, FSA, TiCycles, etc.
Coaching and bike fit - http://source-e.net/ Cyclocross blog - https://crosssports.net/ BJJ instruction - https://ballardbjj.com/

In my experience a break as you describe (at the nipple / nipple head) with spoke still threaded in the nipple is not a result of corrosion, but rather too short a spoke or a spoke with poor nipple insertion. Make sure you check your lengths or tension of other spokes, otherwise the problem might just resurface sooner than you think.

Based on the OP description, I think we're talking about a nipple that broke below the flats when removal was attempted. The spoke is still attached to the section below the flats, which is both seized on the the spoke, and below the level of the outer wall of the rim, and now cannot be unscrewed from the spoke.

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Tech writer/support on this here site. FIST school instructor and certified bike fitter. Formerly at Diamondback Bikes, LeMond Fitness, FSA, TiCycles, etc.
Coaching and bike fit - http://source-e.net/ Cyclocross blog - https://crosssports.net/ BJJ instruction - https://ballardbjj.com/

If I'm understanding your point right, that's 'easily' addressed with a hacksaw, but the issue is mostly how you'd be able to replace the nipple, not the spoke. Which isn't that much of an issue if you can remove the rim tape and you can get to the nipple. Is a cluster if it's a rim that has been designed to be pretty much unserviceable.

Nah,nipple goes in via the opening for the valve stem. Pull it over to the correct spot with a magnetic cable puller. Not a huge deal.

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Tech writer/support on this here site. FIST school instructor and certified bike fitter. Formerly at Diamondback Bikes, LeMond Fitness, FSA, TiCycles, etc.
Coaching and bike fit - http://source-e.net/ Cyclocross blog - https://crosssports.net/ BJJ instruction - https://ballardbjj.com/

Ok. Fair play. Yet another tool/toy that I could need ;-) Must admit that this post is the first time I've heard of that method, although now you say it then it sounds pretty obvious almost. Thankfully I'm tight as a ducks arse, and so all my stuff is still sufficiently old that I don't have any of these new fangled rims.

But in the past I've had a wheel where about half of the nipples seized and then when I was unscrewing snapped off when I was trying to unscrew them with a standard nipple wrench. Thankfully I was able to use a flat bladed screwdriver from the outside to get them off. Blue anodized Mavic Cosmos wheels, still gives me nightmares ;-)

I'm still missing something... Nipples are brass or snondozed aluminum. Neither of which are feromagnetic. How's that work exactly?

 

Oddly enough, we just posted this article here a week or so ago. Check it out!

You stuff the nipple in the latex tube, pull it through. Easy peasy.

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Tech writer/support on this here site. FIST school instructor and certified bike fitter. Formerly at Diamondback Bikes, LeMond Fitness, FSA, TiCycles, etc.
Coaching and bike fit - http://source-e.net/ Cyclocross blog - https://crosssports.net/ BJJ instruction - https://ballardbjj.com/

It’s easier than that. Feed a length of solder through the spoke hole until it reaches the valve hole and pull it out.
Slide the new nipple on (making sure it’s the correct orientation ;-) ), and make a knot behind it.
Pull it back through, and grab with some pliers while you yank the end off the solder.
Get the spoke threaded on and then rattle the rim until the bit of solder falls out.
Job done.

The nipples are aluminum and not magnetic, but you can get the broken piece with some patience out, with some patience, by working it towards the valve hole. If you place a small magnetic screw into the new nipple and drop it into the valve hole you can work it to the nipple hole with a strong magnet.

I'm in the process of replacing all of my nipples and this particular one broke due to corrosion right at the rim wall with the spoke still attached to the interior section of the nipple. As one of the guys suggested, a hacksaw may be the fix needed.