[John]

Since the tangential acceleration is zero, the frictional force must act in the direction of the rod (so it must be angle theta from the horizontal).

The forward friction force on the wheel must be equal to tau_k/r or the wheel would not be spinning at constant speed. The sideways friction force for the friction to act in the correct direction must be equal to
forward friction * tan(90 - theta)
so the total friction force would be
tau_k/r * sqrt(1 + tan(90-theta)^2) = tau_k/r * sqrt(sec(90-theta)^2)
= tau_k/r * sec(90-theta) = tau_k/r * cosec(theta) = tau_k/(r*sin(theta))