The Equipment Kit is in a backpack on a table in the south tower, just to the right as you reach the top of the stairs. Picking it up will give you three Temporary Skills: Arcane Incision, Electrolytic Analysis, and Textossifica Extraction. There is a 15 minute timer on these skills. You can return to the Equipment Kit if they expire before you are done.
The pyrkari to examine are outside the city, both around the base of the walls to the north and well away to the south and west. If you try to examine a corpse using the wrong skill it will result in one of the following, interrupting your examination. If this happens, deal with it make sure you follow the Corpse-To-Skill chart below.
Corpse-To-Skill Chart