Applying your second "trick" and crashing my head against the wall I've got the desired result. However plotting a graph with the resulting function is much faster (0.235151) when I disregard the UM's than when I use them (2.97654). Doing physics calculations with WM can be a nightmare.

Look up Quantity in the documentation:

Quantity[314.159, "Meters"] *
 Sqrt[Quantity[7.34102*10^-6 - 0.0000178801 I,
    "Seconds"^2/"Meters"^2] ]/Quantity[1, "Seconds"]
% // Coth

You can also use your notation, but adding the information that s and m are positive:

(314.159 m *
   Sqrt[((7.34102*10^-6 - 0.0000178801 I)
        s^2)/m^2])/s
Simplify[%, s > 0 && m > 0]