I have been in similar situations in my youth, competing in state wide music competitions, and performing at gigs. In my opinion this is a hard thing to get right because it is not a quality of you and your playing only, but a quality of the relationship between you and the audience.
For the moment let's not discuss what "feeling" really means.
I choose to play a certain way. A way that makes me feel good as a performer. I play the music the way I want (sometimes exactly as written). I play the same way in every performance (at least I think I do). There will be times when the audience is completely drawn in by my playing and I get complements on how much "feeling" I project in the music and other times I get the comment or reaction that I play robotically and have no feeling. When I was younger it killed me, especially since I was trying to get placed in division 1 or pass an audition and I couldn't tell what I was doing wrong (or what I was doing RIGHT). At this point in my like I don't care and I am convinced that this is a subjective assessment that cannot be quantified, and is sometimes used as an excuse to put players in the pass or fail (good or bad) bucket based on whether or not the listener "likes" them. In other words, like I stated before, I think it is about the relationship between the player and the listener and not a thing that any player can replicate for every audience member. There are many players who are constantly put on a pedestal as being great performers and always playing with "feeling" yet they put me to sleep, I hear nothing great and feel absolutely nothing when they play. The opposite occurs too, players who are criticized as not having feeling make me feel! I think it's more on the listener than the player to feel. It is possible that a listener is in a bad mood and simply not open to feeling what the player is trying to project.
Now on to the issue of "feeling". There are several aspects of music performance that can be used to translate feeling to an audience. Several have already been mentioned like dynamics, tempo, etc. But as you pointed out these are indicated in the sheet music. So, in theory if you played the music as written the intended feeling would be there. One way performers project feeling is via body movement and facial expressions. I personally find this very distracting and frankly fake. I'd equate it to being "affected" which is usually a bad thing, like over acting. Music is aural and I listen with my eyes closed, even at a live performance. So the wild body movement and contorted facial expressions are lost on me. I would stay away from a Gestalt approach to describing this as I do not believe is such things. I is possible that this is quantifiable (even thought I indicated otherwise above) but it may be too difficult to quantify as there are too many variables. One thing I see mentioned in answer related to timber. Volume and speed are both part of feeling or help create feeling but the way you attack an instrument changes the harmonic content of the notes. Also, vibrato is very important on instrument that are capable of creating it. On the violin and guitar families of instruments this is one of the most effective ways of adding more feeling to a tune. This device is indicated in the sheet music too but players often add more as they "feel" the need. And when you add vibrato you add the ability to change the speed and force of the vibrato and that really opens up the door to a many variations on tone and sound. Of course it could be overdone and I cannot cite a formula for perfect vibrato. I am not sure how a piano player could emulate this. Based on these comments I would say that I agree with some of the other answers in that you are not expected to play down the sheet music exactly as written but you are expected to apply techniques that help project an interpretation of what is written and that is perceived as feeling.
As a final note I doubt that even a well trained musician could tell the difference between a human player and a very well designed computer simulation that used AI to emulate all these subtle features of a player's technique. In other words "feeling" in music, in the traditional sense, may be slightly a case of self delusion.