The missing compression ring is a sign of trouble and all the wobbling around has probably deformed the headset cups so that unit is no longer trustworthy.
Since a basic Cane Creek, FSA or Ritchey headset runs about $35 you should be able to get a headset, some thin spacers and labor from your LBS to install it for around your $60 budget. Stack height is not a big deal since you can replace one of your 10mm spacers with a 5mm and some 2.5mm spacers to fine tune things and also put a 2.5mm above the stem for better clamping action.