When appling brakes on a 2000 Jeep Grand Cherokee, steeing wheel

iginition time is set too late, ********, readjust.2) spark plugs are fouled or incorrect gapped. 3) distributors contct poits are dirty or pitted.
God bless you

Vibration from about 40-60 mph is usually attributed to wheel balancing, but the onset when you hit a bump seems to indicate play in the steering linkages (tie rods etc) or suspension members and stabiliser bar rubbers but you had that all checked out, that brings us back to the wheel balancing.

1. out of balance tire because it sounds speed spacific.
2. worst of all! Loose lug nuts or possibly loose spindle nut. (you will need to lift the wheel off the ground to tell)
3. worn linkages i.e. ball joints, steering linkage, anything that would allow play in the front wheels.

The part you are looking at is the steering damper.It can be damaged by repeated shaking but won't cause it.
Likely the condition you have is caused by bad lower ball joints.
The lowers have no grease fittings and as such, fail earlier than the uppers. There are other causes but happen less often. Your track bar mount can wear oblong, overall cumulative wear of all front end parts can produce enough side to side play to do that or the wheels may not be torqued down exactly to specifications. In extreme cases where all is good but the shaking continues, you can have additional caster adjustment added to the alignment specs. I am assuming that your tires are properly balanced?

sounds like it time to have the front end balljoints inspected. Only with an inspection will you get the proper results. Why guess, this is a safety issue.

bent wheel, defective tire cord, out of balace tire.

try having the tires balanced. any dirt in the wheels will cause a vibaration due to unbalanced weight or hiting a curb or pothole.

Sounds like the rotors are warped,Take to a brake shop and have them checked or replaced.

have your brake rotors checked they are uneven and warped.Also have the tie rods checked.

Balance the tires. you will find that any tire shop does this very cheap.