<s> (1). Some experts tell me that Laudenbach's paper is incomplete and contains gaps. </s>

I will retract this for now. I do recall being told this, but I am not
aware at this point in time where the gaps in his paper are, if any.  (I do stand by my belief that a number papers in this area are incomplete.)

(2). The result you seek can be  deduced in the following papers by Lizhen Qin (disclaimer: he was my student):

*On moduli spaces and CW structures arising from Morse theory on Hilbert manifolds.* J. Topol. Anal. 2 (2010), no. 4, 469–526

*An application of topological equivalence to Morse theory.* 
arXiv:1102.2838 

In fact, what you ask can be deduced from the first of these papers which handles a metric that is flat near the critical points. The second paper shows that one can actually use any metric such that the function is Morse-Smale.

Alternatively, there is a paper of Burghelea and Haller which also handles the case of a metric that is flat near the critical points. Lizhen tells me that the Burghelea-Haller paper is correct and one can deduce the desired result from their methods.

(The reason I am ranting about this is that I believe this area to be a hotbed of papers containing gaps--with no disrespect to the authors of those papers intended.)