Why did Sherlock Holmes jump from the balcony instead of letting Watson help him in the fight?

It is clearly implied that Holmes faked his death due to unseen factors which might result in forming the next film. It is also a homage, or tribute to the original Author's death and revival of Sherlock Holmes. There might be a bigger or someone sinister going on in the world somewhere and Holmes would like them to think he was dead. Maybe to investigate or do something un-noticed. It is clear, that Holmes planned the whole thing, as he kept the breathing pipe with him from the start of the meeting. He also planned for Watson to witness his death. This shows that Holmes had an ulterior motive at hand when faking his own death.

Sherlock knew that even if he did defeat Moriarty, the evil doctor would never stop hurting him by exploiting his weaknesses, i.e. the people he loves and cares for. Moriarty had already taken Irene, and Watson was going to be next. The only way to stop Moriarty was to kill him right now. The only way to do that was to sacrifice his life.

Holmes was clearly planning ahead to fake his own death and hide, even from Watson. Moriarty's subordinates and organization could easily go after Holmes, even if Moriarty himself was dead.

This stuck in my mind too. Holmes arranged the meeting out there by the drop (he sent the note) knowing he already had the breathing device, so going over the edge was probably on his mind anyway.

In the stories he did it to avoid Moran who he spotted and then he used this to go undercover, he could not let Watson in on the scheme because Watson is unable to be successfully deceiptful (in the stories Holmes will often not tell Watson a plan or lie to him rather than trying to get Watson to put on an act). So it may have been that, Holmes was playing the long game.

Watson was tough, Holmes was tough but injured. Moriarty is tough, possibly armed and fighting like a cornered animal.

However, from a tactical point of view at that point Holmes knew he had Moriarty. The chance of surviving that fall is very low even with the breathing equipment so that chance of Moriarty surviving is not really any higher than if Holmes shot him. If he had let that moment slip Moriarty might have got away, for all we (and Sherlock) knew M had a pistol concealed on him and would have used it had the odds gone against him. Remember from the voice over M was confident that he could outfight Holmes due to Holmes's injury and he exploited that.

In that scene, even if he did take Watson's help - I don't think injured Holmes + Watson would beat Moriarty.

Leaving aside the possibility of Holmes planning to fake his own death, I have an alternate and more Sherlock movie like theory as of why this happened:

I say that Holmes didn't expect Watson to come at the end of their fight and quite possibly - instead had the "area" under the hill (that valley) and the dynamics of his and Moriarty fall figured out (which Moriarty definitely wouldn't have thought about - according to Holmes) just before he made the jump. He also planned the jump in such a way that he would somehow escape the fall (quite possibly by falling into the water, while Moriarty falls on Rocks).

We can see Moriarty screaming all the way down, but Holmes keeping his cool while falling, which mildly indicates this.

Also, by the time Watson comes in, Holmes caught Moriarty in such a way that in a split second Moriarty knew what Holmes was going to do. Given a little more time, Moriarty would have managed to push Holmes down (because he is towards the wall) - So, Holmes didn't have enough time to change his plan - otherwise Moriarty would kill him, and most likely Watson too after that. This is why, he continued with what he planned before.

So, I guess through this scene - Sherlock is shown as more Intuitive and "outside the box" sort of planner when it comes to his "fighting simulations" compared to Moriarty. Clearly, Moriarty didn't expect the possibility of both of them falling down and Sherlock wins!!

Why wouldn't he just let Watson help?

If Holmes and Watson win and Moriarty is taken into custody, it is likely that Moriarty will continue his vengeful pursuit of Watson from prison.

Moriarty: Once we concluded... our business here it important you know... I shall endeavour to find the most creative of endings for the doctor, and his wife.

If Holmes and Watson win and Moriarty is killed, they may find themselves in prison as nothing has yet been proved against Moriarty to the satisfaction of the British state.

Mycroft Holmes: You can't understand the delicacy the situation. I passed the telegram on to my superiors, but they were the ones who have brought Moriarty in to advice on the peace process in the first place. He has positioned himself brilliantly. As one of our formost intellectuals, he's a personal friend of the prime minister... I believe you, but where's your evidence?

Watson: He's too good to leave evidence.... he doesn't leave loose ends.

How could he have survived this fall? Was this part of the plan or just good luck?

Is it possible that Moriarty survived, too?

In summary, Moriarty can only be beaten by a sacrifice play

Moriarty: I... think you've just lost your most valuable piece.

Holmes: A winning strategy sometimes necessitates sacrifice.

Holmes throwing himself and Moriarty over the edge is the final sacrifice play that wins the game.