The kruzman pf install kit comes with a glue kit also. Its an ounce of super glue that is thinned out thinner than water, and 6 Hypodermics with 3 diffrerent size needles. 22ga 25ga and a blunt 1.5 inch needle that is about 20ga (Higher the ga the thinner the needle)
As much as I would like to sell you a kit, they keep the heat/lights on in my shop, I dont think its the best use for this application. When I had to fix delamination like that. I used the needles from the install kit, with wood glue, and then I clamped it down. I dont have a lot of experience with super glue on bonding wood together. It says ca glue is good for wood.
ALTHOUGH, if you use wood glue you have to use gravity, and let it sit clamped for overnight. If you use super glue, then you can do a section at a time and it will stay in place in 10 secs, then move farther down the crack.
I have a thick super glue in the shop that would be best for your situation. I forgot the name of it, but what I like about it is it is colored black and it is thick, which would work good for your situation. With the thicker glue you use plastic nozels that fit on top of the bottle, or you can suck it up in to a pipette, and squirt in there. If there are places where you need to get in there because there is a bubble, I suggest a utility knife, cut with the grain, and then the glue will hold the 2 sliced pieces together.
There is a hobby shop in kalamazoo, that specializes in RC cars and boats, planes ect. they have a great selection of super glue and tips of different size to put on the bottle. I have seen them 2 or 3 inch long, so you cant get in there all of the way.
So as much as I would like to sell you a kit, and I believe you should have one in your tool box, you would be best off buying thick CA glue, with either pipettes to get it in and under the wood, or the specialized nozels that they make for using on top of the bottle. I would suggest a 2 oz bottle of thick, and if you can find the black stuff, even better.