It has been a while since I had thermodynamics and am attempting to solve this problem for work. I need to figure out the other two mass flow rates. I worked it out with specific heats, but don't really trust it. Seems like there should be a simple way to do it with enthalpy; the process just escapes me. Thanks for the help!
With steam in general and absolutely with any phase changes and pressure changes, you need to use enthalpies instead of specific heat. If there are phase changes, specific heat neglects the enthalpy of vaporization. Even if there aren't, the specific heat of steam changes too much with temperature.
You just need to do mass and energy balances (isn't it always mass and energy balances?). The mass balance is $\dot m_h + \dot m_c = \dot m_l$ and the energy balance is $\dot m_h h_h + \dot m_c h_c = \dot m_l h_l$. The enthalpies should be available in any thermodynamics textbook or various software packages.
I'll let you solve the whole thing, but if you have more issues, leave a comment or update the question.