How to fix a broken nipple under the sprinkler head

It depends on where the sprinkler head is located within that zone. If the sprinkler head that you are capping is not a dead end on the line, then you can cap it at this sprinkler head. If it is a dead end, then you need to go to the prior sprinkler head to cap the line.

A dead end occurs with the last sprinkler in a series. Capping a dead end leaves water in the underground line with nowhere to go. If the sprinklers are spaced far apart, this wastes water pressure. In areas prone to freezing, it can cause water in the line to ice and damage the pipe. Dead ends should be capped at the preceding or next-to-last sprinkler head.
Reference: http://www.doityourself.com/stry/how-to-cap-off-a-sprinkler-head

And you do not replace the rotor with a cap but actually just remove the sprinkler head and cap the line. Removing the sprinkler head is very easy to do. Just get a trowel and dig down around the sprinkler head until you have enough clearance to unscrew the current sprinkler head without getting soil in the line.

But before you start the work on capping/removing this sprinkler head, I would try adjusting either the current sprinkler head or the other zones sprinkler head to reduce the amount of water being delivered to that spot. Depending on the type of sprinkler heads you have, you might be able to replace/adjust one or both of the sprinkler nozzles to reduce the flow. Or look into adjusting the pattern on the sprinkler to deliver less water in your area. Or you can even reduce the amount of time that one of these zones is watering. My point is that someone already did all that work to install the line and sprinkler so try some other steps first. There have been plenty of times that I have planted a tree or plant or did some other landscaping that blocked a sprinkler heads spray area and then wished I had another head in that location.

Ok, I'll admit, I'd just buy the sprinkler head. Let someone else do the work of engineering it so that it would work consistently on demand, because this has to work thousands of times, day after day. It must survive variable water pressure, be robust to working in a dirty environment, hard water, heat and cold. Any water system sometimes even has grains of sand, etc., that find their way into the lines. So you must cater to that possibility.

You need a filter. Materials that will be dimensionally stable under temperature variations, without corrosion. A robust design that will not leak, yet survive enough water pressure to spray the water where you need it.

If your goal is to do better than what you would find on the shelf, and this is truly for your own DIY use, then buy it off the shelf, and dissect that unit. Look carefully at it, and decide what makes it work or not work to your satisfaction. I'll admit that I don't think you can make it better for less money than the unit cost you off the shelf. The fact is, you have not said what it was that makes you think the off-the-shelf unit is unsatisfactory to YOU.

If your goal is to try to make something like this better for sale to others, then you are asking in the wrong place anyway. In that case, this question should then be shut down.