one jury rig would be to use what is left of the shifting mech on the hub ,
to put the hub into 2nd gear.

the one in the middle .. that means the cog turns the hubshell directly,
pull it further and its in a reduction gear.. 3/4 of the 2nd gear ratio.


For a freewheel hub any 5 speed freewheel hub will do, you just recenter the axle,
so the cog and chainring line up.

fixie on that bike doesn't make sense.

Of course replacing the broken cable is the most practical thing to do,

Sturmey Archer or Shimano Hub?.

Ask your local bike shop to get the parts.