The coil wire is insulated with thin insulation; your meter won’t show continuity unless you make contact with the very end of the broken wire or remove some insulation.
It looks from the picture that the coil wire to #4 is the outside wire of the coil. IF that is the broken connection, you may be able to unwrap one turn of the coil, remove the insulation and solder it on the tab.
If the coil wire at #3 is broken but there is enough remaining there is a slight possibility that you could solder another wire to the part going to the coil. You need to be careful not to break it again; probably glue the new solder joint to the yellow plastic before soldering the other end to the tab.
IF you are really good at repairs, patient and careful; there is a possibility of taking the steel solenoid housing off and free the coil. This procedure is most likely not worth the effort. The end plate appears to be crimped to the side plates and with a really sharp cold chisel, strong vise and good hammer work could be separated so that it could be re-crimped. With the coil out, it is not too hard to unroll the wire and rewind it. Unroll it on some type of spool so you don't kink it. The tricky part is putting it together so that the plunger operates freely and the joints are strong, it may require a couple of tack welds. It will still function without one or two windings.