Teo has been in the US for only three years, but his teachers say he has learned to speak perfect English in _____ short time because he's so happy at school.
Q. Should I use “so”, “such” or “such a” in this example?
In answer to your question, I am sure somewhere in your text book there will be a note something like this
with singular nouns, we need an article “a or an” before the “adjective” in a sentence.
Therefore "such" cannot be used without the following "a"
The following are the correct terminology however "a" appears in both sentences, albeit in a different places.
learned to speak perfect English in “such a” short time because he's so happy at
learned to speak perfect English in "so" short "a" time because he's so happy at school.
Why has the location of the "a" changed? The answer is simply because the way we use "so short". Short has been modified by "so";
In this sentence we are emphasising the amount of time "so short" = "very, very short"
This link might be useful Using such in English
So + adjective: We often use so when we mean ‘to such a great extent’. With this meaning, "so" is a degree adverb that modifies adjectives and other adverbs:
a determiner (NOT PARTICULAR); used before some words that express a number or amount:
short adjective (TIME): being an amount of time that is less than average or usual:
time noun (PERIOD) A2 [ S or U ]; a particular period of time for which something has been happening, or that is needed for something:
They stayed with us for a short time. = They stayed with us for a short period (of time)
All Ref CED Such