I always get thrown off by mechanisms with just water as a solvent. I've attached my work as far as I could get but I'm not sure where to go from there. Thank you!
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Forget about the water present - its just a solvent. The best nucleophile present is the N-lone pair; if you think about it in 3-D you will see that it can do a nucleophile attack on the CH2-Cl to displace Cl- and form the aziridinium cation. Cl- can then attack this at either carbon of the  ring, one attack merely reforms starting material, the other gives the 3-Cl-piperidine product shown.