Does anyone know how to determine height in Dungeons and Dragons 5th Edition?

If you have the Player's Handbook, check out page 121. It has a chart with four colums, and one row for each race.

The left most number is base height. This is the shortest ( 2 inches, the lowest possible roll) that a member of your race can be and still adventure.

You take the basic height of the character, then roll the dice indicated on the next column to the left. Total those dice up and add them to your basic height. That is how tall your character is.

----

With this situation, it might be a little easier if you convert the feet to inches first. I'll be using the example text you put up to show you what I mean.

Important note: there is 12 inches in 1 foot.

The base height is 4 feet 8 inches.

The feet convert to 48 inches, for a total of 56 inches of height.

The 2d10 roll is 12.

Added to the 56 the total is now 68.

Now divid that by 12 again, do not go into decimal places.

That is 5 with 8 left over.

That means the character's final height is 5'8".

---

In the example's case, it is simpler to convert the dice results first. 12 inches equals one foot. So adding one foot to 4 feet 8 inches, is 5 feet 8 inches.