How tall are the monoliths?
Jelly, (8?) 9 bytes

Ỵ=”|Sḟ0m2


A monadic link accepting a list of characters as specified and returning a list of integers.

Note: 8 bytes if a list of strings, one per line, was really intended to be an allowed input format - just remove the Ỵ.

Try it online!

How?

Ỵ=”|Sḟ0m2 - Link: list of characters, s
Ỵ         - split at newlines
  ”|      - literal '|'
 =        - equals (vectorises)
    S     - sum (vectorises, hence counts the number of '|' in every column)
     ḟ0   - filter out zeros (only keep the results from the sides of the towers)
       m2 - modulo index with 2 (keep only the left side measurements)