Little girl all round the house (5)

I believe the answer is:

villa

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'house' is the definition.
(villa is a kind of house)

'little girl all round' is the wordplay.
'little girl' becomes 'vi' (short for Violet. I am not sure about the 'little' bit.).
'round' is an anagram indicator.
'all' anagrammed gives 'lla'.
'vi'+'lla'='VILLA'

'the' is the link.

Can you help me to learn more?

(Other definitions for villa that I've seen before include "Midlands football team" , "Country residence" , "A place in the country" , "Rented holiday home" , "where 6 [VERDI] lived?" .)

I've seen this clue in The Globe and Mail.

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