You'll need to include the month only if it's not December yet. It's clearer not to include the month when speaking of dates in the same month. Also note that -게요 always includes a sense of intention. It tells the other person about what one wants to do in a friendly tone as if giving them a chance to object if they want. You can compare these two. 1. 저는 21일까지 여기(에) 있을게요 = (If you don't mind) I will stay here until the 21st (cautiously announces one's decision - similar to 있으려고 해요). 2. 저는 21일까지 여기(에) 있을 거예요 = I'll be staying here until the 21st (informs them of already existing arrangements - similar to 있을 예정이에요). -ㄹ 거예요 can sometimes imply intention too, but it more often simply indicates what is to come - it is the future tense from in Korean. I think "I'll be here until the 21st" is closer to (2), so I would say -ㄹ 거예요 (-ㄹ 것이에요) is a better choice.

right. you can omit December. because listener can understand the meaning even though you omitted. ㄹ게요 is okay when you give your words with your willingness. furthermore ㄹ 거예요 is possibly used in the similar situation. so, in normal dialogue you can use 있을게요. 있을 거예요. furthermore ㄹ 거예요 is using when you guess or plan in the future.

You can omit 'December' in the sentence. :)