(1). Some experts tell me that Laudenbach's paper is incomplete and contains gaps.
(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 which also handles the case of a metric that is flat near the critical points. (Lizhen tells me that Burghelea's paper is correct.)
(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.)