Leaky gut usually refers to tissue damage caused by seed lectins, and heals after they're eliminated. Probiotics like kimchi might benefit the gut bacteria but that's a separate issue.
Don't get too distracted by other people's sensitivities and restrictions--eliminating nightshades is not necessary unless they give you distress.
Lots of: urban hiking, cycling, sprinting
Lots of: fresh meat, seafood, eggs, organs, tubers, starch fruits, vegetables, meat fat, dairy fat, oil fruits
Some: cured meat, dairy protein, sweet fruits, rice, pulses, tree nuts, oil seeds
Minimal: soy, refined proteins, sugar, liquid carbohydrate, grains, refined oils, peanuts