With $P$ the diagonals’ intersection and $\theta$ the angle between them, each foot satisfies $PA_1 = PA\cos\theta,$ so $A_1B_1C_1D_1$ is similar to $ABCD$ with ratio $\cos\theta$

Get $\sin\theta$ from area $= \frac{1}{2} d_1 d_2 \sin\theta,$ using Ptolemy for $d_1 d_2$ and Brahmagupta for the area

Let $P = AC \cap BD$ and let $\theta$ be the acute angle between the diagonals. Since $A$ lies on line $AC,$ its foot $A_1$ on $BD$ satisfies $PA_1 = PA\cos\theta,$ landing on the ray of $BD$ making the acute angle with ray $PA;$ the same holds for all four feet. So $A_1B_1C_1D_1$ is the image of $ABCD$ under the map that rotates each ray from $P$ onto the other diagonal (through angle $\theta$) and scales by $\cos\theta:$ corresponding triangles at $P$ are similar with ratio $\cos\theta,$ and every side of $A_1B_1C_1D_1$ is $\cos\theta$ times the corresponding side of $ABCD.$ Hence the perimeter is $(4 + 5 + 6 + 7)\cos\theta = 22\cos\theta.$

By Ptolemy, $AC \cdot BD = 4 \cdot 6 + 5 \cdot 7 = 59.$ By Brahmagupta with $s = 11,$ the area is $\sqrt{7 \cdot 6 \cdot 5 \cdot 4} = 2\sqrt{210}.$ Since the area also equals $\frac{1}{2} \, AC \cdot BD \sin\theta,$ we get $\sin\theta = \frac{4\sqrt{210}}{59},$ so $$\begin{aligned} \cos^2\theta &= 1 - \frac{3360}{3481} = \frac{121}{3481}, \\ \cos\theta &= \frac{11}{59}. \end{aligned}$$

The perimeter is $22 \cdot \frac{11}{59} = \frac{242}{59},$ which is in lowest terms, so $m + n = 242 + 59 = 301.$