$4$

Using the "chain rule" for logarithms which states that $ (\log_a b)(\log_b c) = \log_a c $, we have: \[ \begin{aligned} \frac{\log_59\log_75\log_37}{\log_2\sqrt{6}}+\frac{1}{\log_9\sqrt{6}} & = \frac{\log_37\log_75\log_59}{\log_2\sqrt{6}}+\frac{1}{\log_9\sqrt{6}} \\ & = \frac{\log_39}{\log_2\sqrt{6}}+\frac{1}{\log_9\sqrt{6}} \\ & = \frac{2}{\log_2\sqrt{6}}+\frac{1}{\log_9\sqrt{6}} \\ & = \frac{2}{\frac{1}{\log_{\sqrt{6}} 2}} + \frac{1}{\frac{1}{\log_{\sqrt{6}} 9}} \\ & = 2\log_{\sqrt{6}} 2 + \log_{\sqrt{6}} 9 \\ & = \log_{\sqrt{6}} 4 + \log_{\sqrt{6}} 9 \\ & = \log_{\sqrt{6}} 36 \\ & = \boxed{4} .\end{aligned} \]