Mark, unfortunately you will have to remove the tyre, inner tube and the rubber rim tape to access
(and refit) the lost nipples.
You should be able to do this without removing the wheel, but it will be a bit of a fiddle removing and fitting the tyre and tube assembly with the wheel still in the frame.
With the tyre, tube and rim tape removed, you should be able to persuade the nipples to fall out of the access holes in the rim.
Then you need to get them threaded onto the spokes again. I had to use a pair of fine pointed tweezers to get mine started, but as you have the modified spoke pattern on your wheel there might be an easier way:
If you have a spare spoke, you should be able to screw it into the wrong end of the nipple a couple of turns.
Wrap some sticky tape around the end of the spoke to prevent the nipple from threading on any further, and you should now be able to poke the nipple into the rim and line it up with the spoke and carefully wind it on as far as it will go.
Now, hold the small end of the nipple
(the spoke key end) with one hand, and you should be able to simply unscrew the spoke you have just used as a tool to fit the nipple in place.
Now repeat this procedure for the other nipples and then gradually tighten them until they are approximately as tight as all the rest, whilst checking that the rim still runs true. With your wheel, you may be able to do this with a flat bladed screwdriver, failing that use a spoke key as normal.
As five nipples have already come loose, there's a very good chance that all the other spokes are going to be too loose and will probably need to be tightened up a bit more.
Start with the spoke next to the valve hole and work your way around all of the spokes in turn and try to tighten each of them by 90 degrees
(assuming they will turn that far without getting too stiff).
When you reach the valve hole, re-check the tension of the spokes by either plucking or flicking them. If they don't sound tight enough, go round again until all the spokes are correctly tightened ensuring the rim remains perfectly true. If the spokes are tight enough, the nipples should not come loose, but if you want to put a small blob of locking compound on each spoke/nipple joint it probably won't hurt.
The spokes on my rear wheel
(which I rebuilt over five months ago) have not been touched since the wheel was refitted, and they are still nice and tight and the wheel is still perfectly true
(without any locking compound being used).
Hope this helps.
Alan