The second sentence, 'Mike is going to sing on the night,' can't be used on its own, but it's still correct.
In English, we usually use 'on the night' to describe a specific night. Of the various examples I could use, there are a few basic forms:
Mike is actually a target for an assassination attempt by a terror organization. In all their correspondences between each other, they refer to 'the night,' though it's never stated that 'the night' is when they execute their plan (and Mike.)
In this case, sentences that refer to 'the night' are implicitly assumed to mean that night.
Remember that Mike will be singing on the night. We need to take him out on stage.
If, in a previous sentence, 'the night' is identified, then the value is used by inference. @Billy Kerr provided an excellent example of this in the comments:
The concert will be held on Friday evening. It's going to be wonderful. Mike will sing on the night.
In this case, 'the night' refers to Friday evening.
Extended Literal Specification
Maybe the sentence you provided wasn't complete. If I were to complete it, I'd expand it to something like:
Mike is going to sing on the night of the 25th
This explicitly describes which night Mike will sing on within the scope of the sentence.
This usage could be better viewed as an extension of 'on the (date)'. One would logically say
Mike is going to sing on the 25th
which describes when Mike will sing. If we prepend 'the night of' to 'the 25th', then we talk more specifically about when Mike will sing.
If 'On the Night' is the title of a song, then Mike would be singing that song.
You can't use that grammar without the understanding that 'the night' refers to something specific.