I am not a mechanic but I have done my share of roadside or parkinglot repairs with tempermental distributors. Here are a few suggestions based on my experiences:
1. Disconnect the wire from coil to distributor cap at the cap and with someone trying to start the car, tap the electrical part on the engine block or cylinder head and see if you get a spark. If not, the problem is either in the coil or there is a disconnected/bad wire between coil and switch.
2. If you do get a spark, put the wire back in the distributor cap and take off #1 spark plug wire at the plug. Repeat the test procedure in one, tapping the spark plug wire on the block. You should get a somewhat fast series of sparks reflecting when the rotor hits the #1 contact on the inside of the cap. If you do not get a spark your problem may be the cap as noted:
a. Unlike the Auto-Lite rotor, the Delco-Remy/UMS rotor in the delco-remy distributor has a bent piece of metal. This piece has to make direct contact with the center point on the inside of the cap that the coil wire plugs into. There should be something there to fill the hole in the cap that actually makes contact with the rotor. Often these get bent or burnt (looks like oxidation or "burnt") and the rotor is bad.
b. If you don't see a piece of material in the inside of the cap's center hole, you need to replace the cap. That's the way they did it back then.
c. Check to make sure the cap is properly aligned. You can install the distributor cap 180 degrees off actual location and the timing and firing order get so screwed up the plugs won't fire at the split-second required because the rotor misses the contact point.
d. Check all the small wire bits in the distributor base to make sure they are properly connected and tight. With age, these bits can deteriorate and while look good on the outside, internally the wire strands have broken.
3. Make sure that the broken base pieces align correctly and the distributor itself is properly aligned (see c above for more on this. If you want a new distributor base, Kaiser-Willys Parts in South Carolina had them. You want the base for the 1954 (or includes 1954 model year) 226 engine used in various Willys and Jeep products.
Again I am not a mechanic but having had 226 engined cars for around 40 years before Barbara and I got the Henry J, I had my share of problems and tried to learn about diagnosis and fixes. You might also want to consult the factory service training film on the Electrical system of the 226. While the engine shown is set up as a 1949, the problems and fixes did not change much during the years Kaisers were built. It is on the CIRCLEKF.COM website and when used with the factory shop manual can be a big help.