I am trying to find a way to analyze 2 questions in our survey that are ranked questions.

Is there a way to do this that accurately shows the average weighting so that the top 5 choices are represented?

I don't think there is one best way. It depends a little bit on how much weight you want to give each position.

E.g. you could give points from 5-1 for the first positions and then see the total score or average of each option.

So everytime an option was ranked first you give 5 points, for 2nd 4 points, etc. Or you could say that being number one is worth much more than second or third, so you could say:
1st == 10 points
2nd == 6 points
3rd == 4 points, etc.

As ranking doesn't mean that the difference between 1st and second is the same it is quite risky to create averages. Because the distances between first and second place might be enormous, or they might actually be quite close together.

1....................2.3.4.5
1.2.............3........4.5
1.2.3.4....................5

Those are all totally perfect positions for a ranking from 1-5. So how will you attribute points now for creating an average?

And for each respondent, this scale might actually look totally different. As long as you did not let them position the options on such a scale, but just ask for the positions, creating averages is extremely risky, because it might lead to a totally wrong impression. Therefore, in my opinion, rankings alone are often not very helpful.