Ilse is in the database: German and Dutch diminutive of ELISABETH, however in this case I have another theory (see below).
Ilsing: I found this as a Scandinavian male given name. Also in use as a surname.
There are also places called Ilsingham and Ilsington in Devonshire, England. The name Ilsington is derived from the Domesday name, Ilestintona, which may have developed from the old English personal name, Ielfstan, when the village was first invaded by the Saxons: http://www.ilsingtonparish.co.uk/history.htm
If Ilsington is from Ilestintona, that would mean, that Ilsing is from Ilestin. Ilestin is also used as a surname, no meaning found. Probably there is a relation with the surname Iles, which is of Norman (French) origin. An interesting explanation of the meaning of Iles you can find here: http://www.surnamedb.com/Surname/Iles I would guess that Ilsing means something like "islander". And in this case, the name Ilse would probably not be a variant of Elisabeth, but a form of the described name Iles.