Hello Heather. According to Kelly's 1911 Directory - Milden Cottage (Mrs M Read) was situated on the NE side of Vale Road. Looking through later directories and trying to match the houses on each side, to find out the number, it appears that the "cottage" or villa became No 42, known as 'Sidmount' by, at the latest, 1921.
Have a look using Google Maps and you can see that it is a large detached white house. The area was developed in the early 1900's and the house is contemporary but, I suppose that there's always the possibility that the cottage was knocked down and replaced by a later building.
Constitution Hill, a mile to the west, was known as having the healthiest climate in the area but Vale Road was situated just as near to the Pottery works. Also, the Gas works, at Bourne Valley was quite close by so I'm not so sure that they moved there for health reasons.
By the way, Kelly's 1911 doesn't list any convalescent homes. There was a children's home at St. Faith's, Mount Road, Poole and a Dorset Children's Home (girls) at 32, West Steet, Poole.
Good luck with your research. Sorry that I can't help further but I also do not have access to the 1911 census.
Best wishes
Morten