In a slow cooker, combine the chicken broth, coconut milk, lemongrass, ginger, and kaffir lime leaves. Stir to mix well.
Add the sliced chicken and mushrooms to the slow cooker.
Cover and cook on low for 4-6 hours or on high for 2-3 hours, until the chicken is cooked through.
About 30 minutes before serving, stir in the fish sauce, lime juice, brown sugar, and sliced chili peppers.
Once cooked, remove the lemongrass and ginger slices.
Serve hot, garnished with fresh cilantro and lime wedges on the side.