You need 290 µL of insulin to dose the rat.
Mass of insulin = 2.6 kg rat × (10 mg insulin/1 kg rat) = 26.0 mg insulin
Millilitres of insulin = 26.0 mg insulin × (1 mL insulin/90 mg insulin)
= 0.289 mL insulin
Microlitres of insulin = 0.289 mL insulin × (1000 µL insulin/1 mL insulin)
= 290 µL insulin