Spanish term (edited): Consejería de Sanidad y Dependencia

Based on the definition provided under "References," this agency appears to be a hybrid involving the functions of an advisory body, an administrative department, and a health service delivery unit. 'Bureau' would fit that bill, at least in characterizing agencies in the US. 'Agency' would also work.

As for "dependencia"": the text/context doesn't make it clear what kind of patients are involved. 'Dependency Care' can refer to chemical dependency or care for highly dependent patients. I suspect it's the former. In any case, the phrase preserves the Spanish ambiguity in English. Examples:

Jul 1, 2010 ... Inpatient chemical dependency care rendered in a facility which is not accredited by a nationally recognized accreditation organization ...
www.ncdhhs.gov/.../NCHC-Inpatient-Psych-Care-or-Inpt-Chem-D...

Boyd Andrew Chemical Dependency Care Center company profile in Helena, MT. Our free company profile report for Boyd Andrew Chemical Dependency Care Center ...
www.manta.com/c/.../boyd-andrew-chemical-dependency-care-ce...

Get business info for Outpatient Chemical Dependency Care Center: Alcoholism Information And Treatment Centers in Hutchinson, MN, including address, ...
www.yellowpagecity.com/.../Outpatient Chemical Dependency C...

Jan 25, 2011 ... LGBT Mental Health and Chemical Dependency Care · Follow PRIDE Institute LGBT Treatment on facebook Follow PRIDE Institute LGBT Treatment on ...
pride-institute.com/

Alcohol Dependency Care Homes In Hampshire. » Your search criteria returned 6 results. ... Refine your Alcohol Dependency Care Homes search by location ...
www.carehomestoday.co.uk/0...dependency_care.../hampshire.h...

Other interpretations:

The article investigates four fictional accounts of the complications involved in providing **dependency care for one with a compromised mind**—difficulties the ...
muse.jhu.edu/journals/journal_of_literary.../4.1.marchbanks.html

Empowered Health vs Dependency Care. January 11th, 2011 by Reed Leave a reply ». bigpharma Institutionally-sponsored health care leads to chronic illness and dependency; this can be seen in nearly all senior care facilities where the incentives are for superficially-trained technicians to medicate symptoms rather than do the intensive work of supporting true health..
choosewings.com/?p=858

In spite of the time, energy and resources, material and emotional, social and individual, that dependency care requires, these concerns rarely enter into ...
www.alibris.com/.../The Subject of Care: Feminist Perspecti...

The company also provides "ventilator dependency care"; pediatric nursing services; mental health nursing; home medical equipment and supplies; ...
investing.businessweek.com/research/stocks/private/snapshot.asp?

Nursing actions associated with dependency care The purpose of nursing actions ... There are many nursing actions, related to dependency care, which affect ...
books.google.com/books?isbn=0443104050...

Here's a translation from Spanish that looks nice to me:

Dec 21, 2010 ... Law to Promote Self-Reliance and Dependency Care. ... We have improved and increased the number of dependency care facilities, ...
www.asturias.es/.../21 12 2010 ITV Centro Social mayoresSam...

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Note added at 3 hrs (2011-02-17 00:29:15 GMT)
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Note: 'dependent care' (rather than 'dependency care') would limit the scope to care for one's family dependents, which I don't think is the meaning here.