Why does Walt want to retain Jessie as his partner?

This happens in the episode "One minute":

• Hank has a meltdown and beats the crap out of Jesse.

  

• Jesse ends up in the hospital and is majorly pissed. He wants to destroy Hank through a lawsuit and he also threatens to rat out Walt if he ever gets arrested by the police.

  

• Walt tries to make amends and offers Jesse a job partnership that is worth $1.5 million
  (Walt gets rid of Gale so he can offer Jesse the job)

• Jesse blows Walt off at first, but later calls him and agrees to a 50/50 partnership.

  [Watch a recap of the epiosde]

Walt is clearly a manipulator. He controls most, if not all, of the facets of his life - his wife, her sister, her sister's law enforcement husband. He controls Jesse, his former science partner and his ex-partner's wife, even the school principal for a while. All this took a large part of his time to construct, and a sizeable amount of effort and attention to keep together. I believe that Walt saw Gale as a wild card, something that was to be intrinsic to his existence, but something he didn't design for. Like juggling too many balls, his belief would be that he would not be able to convincingly control Gale; he had to go. But the contract with Gus needed 200 pounds per week, and the lab work required two people for that amount - Jesse remained his partner, not by preference, but by default. The real factor that cemented Walt and Jesse together was the threat to both their lives from Gus, thus requiring elimination of Gale to leave themselves as the defacto meth kings.

An unemployed Jesse is the biggest threat imaginable to Walt (at least in season 3). Walt has an extremely low estimation of Jesse's discipline. This feeling is amplified by Jesse's drug use which Gus points out at their first meeting. After Jesse threatens Walt from the hospital bed (the "get out of jail free card" speech) it's clear that Walt has to either kill Jesse or keep him employed.

At which point the question becomes Gale or Jesse. One of them has to die: Gale, to free the job for Jesse or Jesse, to keep him quiet.

So why kill Gale? By the time it came to the killing there was really no choice but even beforehand you could anticipate it. Gale is almost as good as Walt. So close that it becomes difficult to justify keeping Walt around (especially with someone like Jesse floating around in the background).

Even before the kid killing forced Walt's hand, it was clear: he had to re-employ Jesse whatever the costs.

Yes, this happens in the episode - One Minute. When Walt's wife pursuades him to help Hank, Walt hatches up this plan to replace Gale with Jessie. Walt knew that Jessie was furious over what happened to him and a really big offer was needed to pacify him and let Hank go. Plus if Jessie works with him there is also lesser chances of him getting caught and hence lesser chanes of him giving details about Heisenberg to the police. So this was exactly Walt wanted. This deal served two purposes and hence that was the motivation.

Beside the point that was already made by others (Jesse was a threat to Walt since he could talk to some guys and release Walt's identity), I also think there are two additional minor points why Walt has choosen Jesse:

• Gale was not on Walt's level, but close to it. So after a while there would be no difference between those two. When Walt was working with Jesse it was clear who was "the boss" and Walt liked it to have people arround him who couldn't be as good as him. Walt was having some fun by playing the teacher role in front of Jesse.
• It was boring for Walt to work with Gale. When Walt was asking Gale if he had done a lot of things and everytime, the answer was "yes", he looks kind of disappointed and bored (This was combined with the fact that Walt was searching for errors made by Gale).