Using the definite article "THE" makes the noun more particular or individual. For example, "I finished THE book, means you finished a particular book. Using the indefinite articles "A" or "AN" gives the noun less uniqueness Saying, " I finished A book" could mean any book. For your sentence, it would be normal to say "The day was coming to AN end" since it is a normal thing for all days to end. However, if you were talking about a particular day, perhaps the last day of your vacation, you could say, "I didn't want THE day to end". Hope this helps

"Come to an end" refers more to an event or a process. You would say "Slowly the day was coming to an end," not "the end." The difference is slight, but most of the time you will want to say "come to an end," not the other. A correct example for each: His life has come to an end. He has come to the end of his life. Again, the first is better.