My husband and I want to tqke a week vacation, is there a place in Los Angeles where I can board my mother for a week?

Nursing homes usually do have a couple of beds that they keep open just for this purpose. You should try to contact a few within your area. Good luck on your search.

The problem I found with this is that they will only tell you like a week in advance that they have a bed available, so it is impossible to plan a vacation in advance, will keep you in my prayers

Google: Board and Care Los Angeles or whatever city. There are many places in Pasadena, Alhambra, Claremont area, just plug in your city. Also if Pasadena area call the Huntington Hospital Senior Care Network. Many places do respite care. Call Sunrise there are a number of them in the wider Los Angeles area. Also Google Respite Care and the city name.