I thought that the bond lengths would be different because of the double vs single bonds but it is wrong.

Pragnya do you remember the concept of resonance in unit 2-2.6 | Resonance and Formal Charge.Resonance is the delocalization of pi bonds and lone pairs and molecule true structure is resonance hybrid .In borazine also all pi bonds are moving around or delocalized and the true struture is with partial double bond  between all atoms.That is the reason all bonds in borazine are identical.
See the attached image -Borazine is also called as inorganic benzene .