Temporary (weeks) furnished room (B&B? )

The South American Explorers Clubhouse used to be ideal for this, but has closed as of January 1st.

Getting weekly lodging online without firm dates is problematic for Quito.  Reasonable rates may not be worth posting online by agencies or landlords.

I'd suggest getting a modest-rent hotel room for at least a couple of nights in advance of the trip, and then visiting the Internet cafés for actual (not virtual) bulletin board listings for short-term places.  However, even the majority of these listings will not be by the week.

I visited Quito many times before moving here in 2013.  I used to negotiate a weekly rate at the Ambassador Hotel on 9 de Octubre y Colón.  The negotiation was typically in Spanish and was done with the hotel manager while I was already at the hotel -- not via phone or Internet.

The Ambassador is a sizable, secure but modest hotel within easy walking distance of the Mariscal shops and restaurants.  It is also near desirable transportation.  Nearby are the Columbia barbecue restaurant and the Canadian buffet house, Internet cafés and a bakery, the trole bus line and the Colón Camal, which is a bus line running East-West along Avenida Colón.  There is a restaurant inside the hotel.  Wi-fi was available in the lobby.

cccmedia in Quito