How to determine oxidising and reducing agents in the reaction between ammonia and hypochlorous acid? How do I figure out what is the oxidizing agent and the reducing agent in the following reaction:

$$\ce{\overset{-3}{N}\overset{+1}{H}_3 + \overset{+1}{H}\overset{-2}{O}\overset{+1}{Cl} &lt;=&gt; \overset{-1}{N}\overset{+1}{H}_2\overset{-1}{Cl} + H2O}~?$$

I assigned known oxidation numbers for the elements, but I'm not sure how to proceed with the rest.

Is $\ce{NH3}$ the reducing agent? Or nitrogen? Going from $-3$ to $-1$ 
and $\ce{HOCl}$ the oxidizing agent, chlorine going from $+1$ to $-1$?
You're on the right track with assigning oxidation numbers to each element. And you've assigned them successfully. 

To successfully complete the task you need to define a few terms:

What's oxidation?

What's reduction?

What's the oxidizing agent? Try thinking of this in terms of what oxidation is, and then try figuring out what an oxidizing agent would do. 

What's the reducing agent?