Since the meeting took place at a definite moment in the past, the verb should be in the simple past tense ("met") and not in the past perfect. The following verb should also be in the simple past -- and in the active, not passive, voice ("decided"). Finally, because the centre would probably benefit the current residents, not those who were living there previously, "had been living" should be changed to "are living" -- or, even better, simply "live".
2. Noun-pronoun agreement:
"Committee" is a collective noun, so unless you want to emphasize the individual members of the group, you should use the singular pronoun ("it").
3. Phrasing error:
You don't want to say that they decided something that already happened; they decided on a future course of action. Use the infinite to indicate the action of the main verb ("to approve"). Likewise, use the infinitive to indicate the purpose of the main verb ("to benefit")
4. Word form:
The committee approved something, so we need a noun. This may be a little tricky since "constructing" is a gerund (which also functions like a noun). But in this case, we should use "construction" (or possibly "the construction"). Also, we need a noun form to modify "centre" ("retirement") rather than the present participial adjective ("retiring").
5. Word confusion:
"There" is an adverb of place; here you need the possessive adjective ("their").
6. Plural form:
The plural of "family" is "families". Do not confuse it with the possessive ("family's").
7. Relative pronoun:
When referring to people, you should use "who" not "which".
8. Article use:
When referring to something definite, use the definite article ("the"), not the indefinite one ("a"): "the surrounding area". Also, since we know precisely which seniors (and families) will benefit, we can add the definite article there as well: "the seniors and their families".
"Seniors" is a not a proper noun, so it should not be capitalized.
After an introductory phrase or clause, use a comma.
When the committee met last week, it decided to approve construction of a new retirement centre to benefit the seniors and their families who live in the surrounding area.