I'm a microbiologist in dairy.
Only two things normally grow in such an acidic environment as kimchi. Yeast or fungi (mold).
The two microorganisms are practically the same. Fungus is a mutiple-cell organism and yeast a single-cell organism.
The two can change under influence of circumstances (such as temperature). So fungus can become yeast and the other way around.
Normally fungus can make harmfull poison. Yeast normally does not.
Both organisms are living from the same source in kimchi, called sugar.
As kimchi should ferment because of lactic acids (bacteria) the sugar should be eaten by those lactic acid bacteria. Whenever you add sugar later or not all sugar is fermented well, other organism can grow in kimchi such as yeast.
If one puts too much sugar in kimchi there will remain sugar before the lactic acid bacteria change the sugar into acid.
Normally lactic acid bacteria die because of too much acid. They should die because of their own product.
A small a mount of oxygen could be very harmful and cause other organisms to grow when there is sugar remaning.
So how can you be sure? Fungus is normally shown only on the top side of the kimchi and makes an unclear liquid. The kimchi liquid should be clouded.
When yeast infected the kimchi, the kimchi should be harmless. The smell shold be weakend because of the growth of yeast. There should be a small alcohol smell, but that's hard to notice. The kimchi liquid should be clear.
I also need to inform that fungi (mold) make poison. The fungi itself are normally not harmful. So the danger is the poison, heating will not change the poison and is no solution. Always be carefull with it, but if you are sure there is alcohol smell in there, then it should be safe.
By the way thanks, I do have the same problem over here after putting radish in my 6 months old kimchi. I thought the kimchi shold preserve the radish, but, I guess the radish contains some sugar and natural yeast. So exposed to a small amount of air the yeast started to grow.
How do I know it is yeast? My white spots are on the bottom of the jar.
My lactic acid bacteria already died, then I put the radish in. So the radish went bad and the lactic acids weren't there anymore to prevent that. I did throw it away because I do not feel safe.