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A Kinetic Blast is an SPA - Spell-Like Ability. You only get one spell per round and this amulet will not change that. (Yes, I know about the Quicken Spell metamagic feat, but that only applies to spells, not SPAs.)
Note that the description of Kinetic Blade is very specific about it being a blast, despite being used in melee.
This amulet would help out the user of Kinetic Whip, however, as KW is quit specific about it being able to perform Attacks of Opportunity. (KW is presented as being able to make more than one attack per round without explanation of why it isn't limited like Kinetic Blade; it just does.)
So in the end, it depends on whether your Kineticist uses Kinetic Whip or not.
An important thing to note though: Even though it technically doesnt work, theres no reason you cant allow it to work in your game. Maybe even switch the restoring ki aspect to absorbing one point of burn instead
As far as I know, kineticists dont have a lot of official item support. There's the kineticist diadem and overflowing rod but those are both out of the price range you listed.
A conductive weapon could be a good prize as well to help deal with spell resistance (I think that works, if Im wrong someone will correct me).
If fire resistance is coming up a lot in the campaign I could see a 9k item doing something like halving the damage on the blast in order to get through the energy resist.
N Jollys guild has a solid breakdown of some magic items including 3rd party items if you want to use or adapt some of those.
