The area of a room is

Area of a room $=$ Length $\times$ Breadth Thus, we have:

$65 \frac{1}{4}=$ Length $\times 5 \frac{7}{16}$

Length $=65 \frac{1}{4} \div 5 \frac{7}{16}$

$=\left(65+\frac{1}{4}\right) \div\left(5+\frac{7}{16}\right)$

$=\frac{261}{4} \div \frac{87}{16}$

$=\frac{261}{4} \times \frac{16}{87}$

$=\frac{261 \times 16}{4 \times 87}$

$=\frac{4176}{348}$

$=12 \mathrm{~m}$

Hence, the length of the room is 12 metres.