I will help you out, because this was me just a few months ago until I sat down and did my homework.
From what I have read, your Deadlift workout would look something like this:
Just from this, sirens can be heard in the distance.
Lets start with the PR set. You say you go "balls out for a PR and it all just becomes a messy touch n go grind". The PR set should not be grinded out, nor should it be taken until failure. Each rep should be done with maximum effort and speed until you have 1-2 good reps left in the tank. From this, I can safely assume that your TM is set too high. Make sure that it is 85% or BELOW for deadlifts.
Jokers. I'm not sure there are many people on this site who understand Jokers, and by no means do I consider myself an expert at them. Jim himself has admitted that he sometimes wishes he had just kept his mouth shut about them. Anyway, I do know this: Jokers are NOT part of your daily workouts. They are reserved for those very special unicorn days in which your weights feel like styrofoam. Until then, take them out.
FSL and your 10x10. You have to pick one. I'm not even sure if the 10x10 is even advised for 5/3/1. In the Beyond book, Jim mentions GVT, but disses it at the same time. This leaves you with FSL.
FSL, BBS, BBB, Widowmakers, Pyramid, whatever other template you see. That is your supplemental work. You pick ONE of these to do, and then move onto assistance.
Assistance work is abs, low back, triceps, curls, lat pulldowns, kroc rows, dips, DB bench. The list goes on. On your upper body days you will pick: rowing movement, pressing movement, "core" movement (abs, low back, single leg work). The rule of thumb for these lifts is 50-100 lifts.
They you will do your conditioning work (with jumps/throws/sprints preceding the workout).
So lets modify this Deadlift workout:
3x5 Box Jumps
5/3/1+ (85% TM, 1-2 GOOD reps left in the tank)
FSL 5x5 squat or deadlift just use the appropriate weight
Leg Raises 50-100 reps
20 min Air Dyne
Hope this helps. I will try to answer more questions if I can.