Which are correct:
1-A hospitalisation of how long is necessary for the patient to recover completely?
2-A stay of how long is normally necessary for the trainee to get acquainted with the workplace?
Original Post
I don't consider either of them too frequent. Compare:

16 on "how long of a stay".
http://books.google.com/books?...22&btnG=Search+Books

2 on "a stay of how long"
http://books.google.com/books?...2+&btnG=Search+Books

and of these two, only this is relevant and idiomatic:
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Parliamentary papers‎ - Page 139
by Great Britain. Parliament. House of Commons - History - 1926

That was 22 guineas for a stay of how long — three weeks or a month ? — She
stayed over six weeks. ...
-----

I'd say try instead:

How long (of) a hospitalization is necessary ...
How long (of) a stay is necessary ...
quote:
1-A hospitalisation of how long is necessary for the patient to recover completely?
2-A stay of how long is normally necessary for the trainee to get acquainted with the workplace?

Both sentences are correct.

These are also correct:

1) How long a hospitalization is necessary for the patient to recover completely?

2) How long a stay is normally necessary for...
_______

I would not use 'of' after 'how long.'

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