Zacspace

Your flyby trajectory is pretty extreme. The burn from low Kerbin orbit to make your Jool transfer should only cost you 1900-2100m/s. Ideally you want to come in ahead of Jool travelling almost the same direction and then get scooped up by it's gravity. As it is you're kind of almost smacking straight into the side of its SOI as it flies by.
As is, your options are to try and get a beneficial gravity assist off of Tylo, or try and line up an aerobrake in Laythe's upper atmosphere. I think it might take some combination of both.
As for what you should send, or anything else related to vessel construction, we'd need to see your ship. That said fuel is a good bet, or heat shields if you're going to go the aerobraking route. You might want to consider dropping your science equipment and using the landers as drop tanks. Better to send a rescue mission to Jool than to high solar orbit, even if you don't end up getting to do your planned mission.
Edit: You're trying to brake into Laythe orbit all in one go while descending straight into it. For starters, you'll want to make a course correction burn in solar orbit. If you play with the prograde/retrograde and normal/antinormal nodes you should be able to adjust the timing of your arrival so that your Laythe entry trajectory isn't so suicidal. This is also how you'd go about lining up a gravity assist off of Tylo. For best results at Laythe, meet it on the right side of Jool and enter the atmosphere on the right side of Laythe. That way you'll have the lowest relative velocity to Laythe, Aerobraking should be more survivable and your burn should be more reasonable.

Your flyby trajectory is pretty extreme. The burn from low Kerbin orbit to make your Jool transfer should only cost you 1900-2100m/s. Ideally you want to come in ahead of Jool travelling almost the same direction and then get scooped up by it's gravity. As it is you're kind of almost smacking straight into the side of its SOI as it flies by.
As is, your options are to try and get a beneficial gravity assist off of Tylo, or try and line up an aerobrake in Laythe's upper atmosphere. I think it might take some combination of both.
As for what you should send, or anything else related to vessel construction, we'd need to see your ship. That said fuel is a good bet, or heat shields if you're going to go the aerobraking route. You might want to consider dropping your science equipment and using the landers as drop tanks. Better to send a rescue mission to Jool than to high solar orbit, even if you don't end up getting to do your planned mission.
Edit: You're trying to brake into Laythe orbit all in one go while descending straight into it. For starters, you'll want to make a course correction burn in solar orbit. If you play with the prograde/retrograde and normal/antinormal nodes you should be able to adjust the timing of your arrival so that your Laythe entry trajectory isn't so suicidal. This is also how you'd go about lining up a gravity assist off of Tylo. For best results at Laythe, meet it on the right side of Jool and enter the atmosphere on the right side of Laythe. That way you'll have the lowest relative velocity to Laythe, Aerobraking should be more survivable and your burn should be more reasonable.