I went where you think you need to go.
It takes more than just the reseal kit - 04445-60050 - Steering box reseal kit
To completely take care of it you also need the following:
2* - 44154-30020 - Power piston plunger guide nut - Partsouq - $5 - I required 2 of these because I went just with the reseal kit at first, then did 44120-60180 below. These are very shallow hex-head bolts and the heads are prone to stripping out easily.
1 - 18-32-7-DL - Input shaft seal part number (not Toyota) - Ebay - $8 - This part may or may not come in 04445-60050 above, it did not come in my kit. This is the P/N in case you need to order it separately. This is the bottom seal for the shaft, iirc.
1 - 44120-60180 - VALVE ASSY, W/BALL NUT - TPD - $433 - This is the part that resolves the play in the steering. It replaces the valve assy (for which there are no replacement seals available), the ball set, the 'worm screw', and the power piston. The 7 - 10 degrees of play comes from the wear between the balls and screw. Some places can replace the balls with larger ones, but total cost is about the same between a DIY with these parts and sending it off somewhere.
Oh and don't forget a few quarts of Toyota ATF. Think I burned through about 3 quarts trying the cheaper fix first approach. So messy and a waste of time until you get 44120-60180 replaced.