HMMT November 2021 U21

Circle $\omega$ is inscribed in rhombus $H M_{1} M_{2} T$ so that $\omega$ is tangent to $\overline{H M_{1}}$ at $A, \overline{M_{1} M_{2}}$ at $I, \overline{M_{2} T}$ at $M$, and $\overline{T H}$ at $E$. Given that the area of $H M_{1} M_{2} T$ is 1440 and the area of $E M T$ is 405 , find the area of $A I M E$.