From what your discribing I'm pretty sure that what your seeing is what we call an Areomonas infection which has caused your koi to end p this way.
As the water starts to warm up these bacteria can multiply rapidly causing sores then ulcers to your koi .
One thing you could do in future is to prepare your pond and filters for the wnter period .
If you dont have a bottom drain on your pond you'll need to give the bottom of the pond a damn good clean removing all dead leaves etc from the bottom of the pond.
In the late fall we clean our filters then through the winter months we carry on with partial water changes to our pond using the trickle method, this allows the water coming through the hosepipe to take on the ambient temperature of the water in yur pond, if done correctly you wont loose much say .1c .
To Keep our water fluid we use Policarbonate roofing to protect the pond from the worst winter can throw at us all our pipe work is lagged and he filters are housed in their own little shed like sructure with the policarbonate as a flat roof and the sides of the walls insulated with 22mm polystyrene sheets like wise our pond which is free standing .
Then we wrapped the polystyrene in silversided bubble wrap and finished off with re-cycled plastic loft insulation this is then cladd in decking , the result , even at our worst winter temperature we dont freeze .
We saw -15c but the pond was at a constant 7c throughout non heated.
We have however placed a small oil heated radiator in the filter housing should we wish to bring the temperature up , believe it or not when I say its cheaper than running an inline koi heater of 1kw .
Back to your areomonas problem the bactria that causes this is called Areomonas hydrophilia which can be caused by a numbe of factors such as overcrowding, poor water conditions, temerature fluctuations plus other factors that have the effect of over stressing your koi so that the bactia which is always in your pond but isnt normaly a problem becomes a major one.
We first need to identify which bacteria is causing the poroblem as Flavobacterium Columnare can be another bacteia that can cause a problem.
So your first action is to take a swab to send to the vet to identify which one your up against.
If caught early you can normally buy a good off the shelf bacteriacide
But it sounds as though you have areomonas hydrophilia from the reddening on the body and build up of mucus, this in turn can lead to scale lifting and ultimately dropsy and ulcerations on the body
As I have mentioned a good Bactriacide should do the trick if as your dscribing weve caught this early enough.
Prevention I have already mentioned is a clean enviroment, not over crowding your koi etc
Adding a UV sterilizer to your pond will efectively nuke any bacteria in the pond but the best method of prevention is to clean your filter in late spring mid summer and late fall along with water changes if possible and your going to get koi like our own "when the covers came off they were in immaculate condition .
Koi keepers in the know call this period of the year "Areomonas Alley", which is when the bacteria attack your koi through as already mentioned , when the koi are at their weakest point .
Parasites also start to take advantage of your koi after winter so I would suggest when the temperature is warm enough to feed your koi buying Wheatgerm and Garlic pelleted food , if you cant buy these pellets add garlic extract to your wheatgem pellets as parasites do not like the taste and as such it gives your koi rudimentary protection against them....
But abov all practice good husbandry of the pond and your filters and this sholdnt ever be a problem to you again
Dave