As retired heavy truck technician, I replaced a fair amount of kingpins in my time. Beside getting everything apart, the main thing to check is kingpin fit in the axle eye and excessive wear on the thrust areas between the face inside the spindle and the face of the axle eye where the shims washers and thrust bearing are supported. Most kingpin kits only come with up to 4 thrust washers. If excessive were is noted in these areas, then a special hardened thicker shim may be need to be installed to control the up and down play of the spindle. Before taking the front axle apart, check for wheel bearing play by tightening the wheel bearings to see if your play goes away, tighten up the wedge bolt in the axle eye to see if the king pin is actually loose in the axle eye and with the axle on jack stands, use a heavy prey bar and block of wood to see how much up and down play you have. This will let you know what to possibly expect when you take it all apart.
If possible, have your kingpins fitted in the new bushings at a machine shop that can line hone them. Using a piloted reamer will work but a better fit is the outcome of line honing. Before installing the spindle on the axle eye, touch up the thrust surfaces with a file to get rid of any nicks or uneven areas that will give you a false reading when checking the thickness of the thrust washers to be installed. When assembling the spindle back on the axle, installed the king pin and thrust bearing dry. Then place a small bottle just or floor jack under the bottom end of the spindle with a block of wood. Apply enough pressure to just barely lift the axle off the jack stand. This will simulate the load on the thrust bearing. Shim the area between the top axle eye face and lower spindle eye face to get the correct thrust. Do not get too tight or the grease caps will blow off when greasing the assembly after it is assembled. Usually you try putting shims in until they are just snug and then leave one out.
Back in the 80s when I first started working on heavy trucks, it was common for an axle with a single wedge pin to slightly wobble out. We would heat the axle eye up red hot with a torch and then hammer it down to shrink it and then let it cool slowly. Then use an adjustable reamer to fit the new king pin. This has now been illegal to do for a number of years because of a safety issue.
If you need king pins for your 31 Chevy, I believe i have standard, .005 and .010 oversize kits in stock. These are NORS king pin sets.
