What plants would benefit from a top dressing of wood ashes?
You need to get a soil test to see if your soil needs potassium before even considering adding wood ashes. Also consider that wood ashes will raise the soil pH. In many cases, people are struggling to lower the pH, not raise it. Plants will not grow well with a raised pH. It will take years to lower it if you raised it high enough.
If you take wood ashes and mix them with water, you have the lye that was used by Grandma to make lye soap. In other words, it is very caustic. Water on wood ashes that are applied to the ground can injure or kill plants if too much is applied.
You can get a soil test to find out about the potassium levels, but there is no way of knowing what the amount of potassium is in the ashes. Different kinds of trees were burned in different amounts. It's one thing if the bag of 0-0-60 has a label and you have a bucket of wood ashes with no directions. It is probably best to mix them with your compost pile so they are not as concentrated when used.
Would my county SWCD have testing kits or where might I get one?
Go to http://soiltest.cfans.umn.edu/ for kits and how to.