Hey, the number of wires in the schematic doesn't match the pic.
Take another pic, and show the rear of the switch. Make the wires perpendicular to the back of the switch, if you can.
The white wire jumper seems out of place. Using the rear punch downs and the screws is sometimes not allowed. One or the other.
If things were "pigtailed" like they are supposed to be, this would be easier.
It looks like you have a neutral in the switchbox, which doesn't happen in my house.
Confirm that no light is controlled from two different locations.
Do you have a test light and or multimeter?
The wiring needs to be first converted to a pigtail. This will also be safer.
Since your new, I'd like to point this
https://www.make-my-own-house.com/light-switch-wiring.html page out. Note, the white wire is coded black = This is a situation you don't have. None of the wires in the switches are coded black in my house. Coding black means wrap the end of the insulation with black electrical tape. Now you know what the colored electrical tape is used for.
YOU need to take one of the switches and "CONVERT" it to a "PIGTAIL". Ideally, there should be one black wire exiting the switch and it heads toward the lamp. If there are multiple lamps, then there could be multiple wires. Then there could be a bunch of black wires connected to the other terminal of the switch and these will be always hot.
So, for one, you need to find what end of the switch you can TOTALLY disconnect and still have power to your kitchen. Then take all of the wires on the other side and wire nut them together and make a pigtail. This will be your always hot lead.
My best guess, is the side of the switch with most of the wires is HOT all the time.
The other side of the switch, if it has multiple wires, it may go to multiple lights, like two or so.
So, some critical things:
1) the switches have only two screws? Assume yes
2) How many wires go to the top screw or back of the switch at the same terminal? Color? (assume black)
3) How many wires go to the bottom screw or back of the switch at the same terminal? Color? (black)
4) What does that lonely white wire do? Should it have been coded black? That's my guess. Basically, it's a hot wire, I think.
5) All of the poked in wires going to the same jumped location is probably hot all the time. One or two pigtails should have been created here. If one, then the switch could have been wired daisy chained using the holes in the back. One wire to a screw terminal.
Getting the timer and all of the wires to fit in the box is going to be tough.
There are some useful wiring devices, but I'm not sure you'll need them. One has a wirenut and a wire that comes out of the normally closed end. I like them a lot.
There is another thing I don't like about that install: Newer switches should have a ground lug. I'm not sure it's present. You are NOT allowed to rely on the screw to make the ground connection to the box, It MUST be done with a wire. (Aside: There are special switches that can be installed without the wire)
I need a better pic from the rear of the switch, preferably angled slightly so the side screws are seen.