As W@nted says, it is not advisable to disassemble a Sti lever.
I had to do a bit of disassembly because like the OP a gear cable end (slug?) was stuck inside, and I too am using ST 6600 which does not have the removable bottom plates provided in newer brifters [brake/shifter].
Once I had removed the name plate (one strew), the black top cover (another screw and the front nut with lock tabs which I managed to ignore by using some radio pliers), side covers (one interior screw joining the two sides together) and the brifter base (remove a grub screw with a 2mm Allen key from the bottom and push the pin/axle of the brake lever out), I managed to hook the broken end of the cable out (on the inboard side) using a hook fashioned out of a braddle, the end of which I had bent.
I am not sure which springs W@nted is referring to. I did not take apart the main mechanism springs of the STI itself.
But the most difficulty part for me was simply (?!) reassembling the brifter base, due to the the spring that keeps the brake lever open. This small round screw about 10mm in diameter fits onto an outer axle inside the front part of the brifter. One of the spring coils ends is bent and this fits into a hole in the brifter cage. It is the other, straight end that presents a problem.
A video on YouTube explains that one is meant to have a special tensioning tool to ensure that the spring is in tension when replacing the base of the brifter. The tool is just a narrow gauge tube that fits over the end of the straight part of the spring to extend it so that it engages with the base.
I do not have the special tool but I managed to use a football inflation needle and put that over the straight end of the spring.
Here is the video. No affiliation/connection at all. It should start at 8:19 which is where the videographer explains how to tension the spring with a special tool.
