The fact that a separable Banach space is a quotient of [tex]\ell_1(\mathbb{N})[/tex] is pretty well-known, and a proof can be found for example on pages 103-104 of
Morrison, Functional Analysis: An Introduction to Banach Space Theory. Wiley-Interscience, 2000.

I just scanned the proof and I think you can easily modify it to get the result you want. Or you can try Lindenstrauss-Tzafriri, Dunford-Schwarz or Megginson to see if they have a proof of the general result.