Can I just ask at what speed I should apply the hand break ...
Any manoeuvre of this type must be done with safety in mind. I suggest applying the hand brake at no more than 30mph on local roads (not a motorway) and with as few other vehicles around (moving or stationary) as possible. Again this is to reduce risks.
The brakes will only "bed in" if the car is moving, and as you are applying the (rather weak) handbrake, you can apply the throttle to keep the speed at 30mph. But if you are not keen to apply the throttle, then allow the car to slow to say 25mph, release the brake, speed up to 30mph and apply the brake again.
The only reason I suggest holding the handbrake button in is to make it easy to release the brake. There is no great science behind the process, it's more a case of how confident you are at controlling the car safely while using the brakes in this unconventional way.
There are two things going on - 1) bedding in the pads which causes them to wear. This results in greater movement on the hand brake, and 2) Rotation of the automatic hand brake adjuster to reduce the hand brake movement.
There is no doubt that these processes will help the brakes adjust, but the questions are by how much and will the effects be noticeable.
I forget which side wasn't bedding in so well. If it was the side with the old calliper, there could be rust or other debris causing the pads to sit at an angle to the surface of the disc. The calliper slides could be worn or seized. If it is the old calliper side, then all the mating surfaces should be checked for any rust build-up. Any present needs removing, and the slide pins should be refurbished as necessary.
This all comes down to the process of fitting. This is a particular skill. To a large extent it needs teaching. It would take a lifetime to learn without help. So it really depends how experiences both you and your friend are. Working at Honda is not necessarily a qualification as a fitter. Being able to "fix a pile of parts together", likewise is not necessarily that qualification. Fitting is looking, touching, feeling and measuring to see how parts mate and move together; and then making adjustments so that mechanisms "operate properly" - whatever that means for the mechanism. A half-worn disc shows that the something is not square and that can be caused by many things, some of which I have just mentioned.
HTH