200 mW is plenty to power up most modern white LEDs. I would likely not pick one that's designed for many times that, but anything up to a few watts should work.
The characteristic that you seem to be interested in is efficacy. It's how much light output you get for your power input. It is usually stated as lumens per watt, or more specifically lumens output per watt input. This is similar to but not quite the same as efficiency, which would be watts output per watts input. The difference is another whole discussion.
The efficacy of LEDs, especially white ones, is increasing on a regular basis, so you are likely to find that for any given manufacturer, the most recent releases tend to have the highest efficacy. Also, efficacy is dependent on the drive level. For any given LED, the efficacy will peak at a specific drive level. This is usually several times less than the 'rated' drive, which is usually several times less than the 'max' drive level.
A white LED operating well below its rated current will likely have color/CRI shifts, but that doesn't seem important to you, so you have a lot of options.
Since you are looking for high efficacy at 200 mW, you are most likely looking for LEDs rated for around 1W. Max ratings might be a few times that.
To give you the best and most specific advice, we would need to know some more about what you want. Beam pattern, optics type, size restrictions, your skills and knowledge, are all important. The more you can tell us the more we can help you.
BTW, welcome to the forum!