I may recommend a guest family.I am staying with a family but thy only have one room available and I am staying with them.
There is a lady who taught me Sapnish for a while.
She has bought a beautiful flat (old style) and she told meshe is planning to rent .It is not a hotel but it is a kind of Bed and breakfast. It is very close to the downtown area.
I can not tell about prices but I believe they are lower than what you were expecting.
There are boutique hotels too especially in Palermo they may be in range of prices you mentioned.
asegards weather it may be cold or rather warm too that is unpredictable.
The main attractions are the ones mentioned above.There is also a wonderful Sunday fair in mataderosan area somewhat distant from downtown but you can not miss it.
If you want interesting historical tours to get the most of the city you can contact a goup of historians of thelocal University (UBA) their mail firstname.lastname@example.org
if you happen to be interested in renting the bedroom I mentioned their email is email@example.com
I do not remember if it was mentioned but Tigre (the delta of the river Parana is a place you can ot miss. It is an hour ride by train from Recoleta.
You can also visit a gaucho town called San Antonio de Areco (100km from Buenos Aires) buses take 2 hours to get there from Retiro bus station.
I hope the information has been useful