There isn't enough context and the line could be ambiguous. But let's just assume this is a thought of someone who is talking about themself planning to do the action described here.
This sentence final particle (終助詞) か is explained here:
It works to convey your intention of doing something, but compared to the version without か in a way it's almost like tentatively running your idea by the listener and seeking their agreement. Here as described in your context, there is no listener, and the speaker is talking to themself. Without か the sentence sounds like the speaker is just stating a fact:
Now that I feel refreshed, I will leave for the military headquarters.
With か, on the other hand, the statement is softened and sounds more natural when one states one's intentions, when to oneself. The effect it brings to the line is almost like tag questions
Now that I feel refreshed, I think I will leave for the military headquarters.
Now that I feel refreshed, I shall leave for the military headquarters, shan't I.
Now that I feel refreshed, let's leave for the military headquarters.