Let's make the lines of the story:
- 他想敲碎這面鏡子,可是這鏡子太高了,"所以他找了ㄧ把椅子站在上面...". The last phrase can also be constructed as "所以他站在ㄧ把找來的椅子上面...".
Note that both contain the expression in your thought - "he stood on the chair". And, in both, 把 is used as a "measure word(量词)", similar to "ㄧ張", "ㄧ個".....
Now, let's see what 把 means/does in the pattern of 他把...
他把事情搞砸了 = He failed the thing.
他把我搞糊塗了 = He confused me, or He made me confusing.
他把椅子放在鏡子前 = He placed the chair in front of the mirror, ...
In the examples above, 把 always associates with an action (搞 and 放 respectively) that causes/triggers an effect (砸了,糊塗了,在..前) on the object (事情,我,椅子) following it. The full sentence of the last example can be:
- 他把椅子放在鏡子前,然後站在那把椅子上面. This includes both the pattern "他把" and the measure word "那(一)把", as described in the beginning.
What is wrong with 他把椅子站在上面?
Grammatically no mistake, but it does not reflect the thought you wanted to express - "He stood on the chair". Instead, it vaguely says "He placed the chair on top" (on top of what?), which can be improved by adding an object after 在 - "他把椅子站在桌子上面", or adding an action - "他把椅子放好站在上面 (He stably placed the chair and stood on it).