The Standard Model (SM) has proven to be excellent in describing and predicting the phenomena studied in High Energy Physics (HEP), but there are some experimental evidences that are still not explained. New Physics (NP) models have been developed and they are tested in HEP experiments at accelerator such as the Large Hadron Collider (LHC). LHC collides protons almost head-on at a nominal center-of-mass energy of $\sqrt{s}=14$ TeV ($\sqrt{s}=7$ TeV during 2011 and $\sqrt{s}=8$ TeV during 2012). One of the four main detectors at LHC is LHCb, with a physics programme originally focused on $b$ and $c$ quark physics, CP violating processes and rare decays, intended for indirect searches for NP as unexpected contribution arising in loop diagrams. The flavour-changing neutral-current (FCNC) \Ksmm{} decay is also a rare decay whose search has been proposed as a novel way to profit from the excellent performance of the LHCb detector. The non observation of the \Ksmm{} decay played an historical role in particle physics, inspiring the GIM mechanism which predicts the suppression of FCNC decay with the introduction of a new quark, later identified as the $c$ quark and observed with the discovery of the $J/\psi$ meson. The \Ksmm{} amplitude in the SM is well predicted and evidence of \Ksmm{} well above the SM prediction ($(5.1 \pm 1.5) \cdot 10^{-12}$) would be a clear signal of NP, while the non observation of such enhancement will provide significant constraints on several consistent NP scenarios. The search for \Ksmm{} is challenging at LHCb because the LHCb detector is optimised for $B$ mesons resulting, in the \Ks{} case, in a lower reconstruction and trigger performances that have to be taken in account. However, an analysis of the \Ksmm{} decay was performed with the 2011 LHCb data set, yielding an upper limit \BKsmm{} $< 11.2(9.0) \cdot 10^{-9}$ at 95 (90) \% confidence level which improved the previous limit, set at the CERN PS in the 1973, by a factor of about 30. In this thesis, a contribution to the upgrade \Ksmm{} analysis at LHCb, using the not yet analyzed 2 $\mathrm{fb}^{-1}$ data acquired in 2012, has been given and the potential improvement of the search sensitivity has been estimated. My original contributions to the analysis included the trigger studies, for the development of the Boosted Decision Tree (BDT) to reduce the combinatorial background, and for the optimization of the muon identification requirement to reject the misidentified \Kspp{} events that contribute to the background in the signal region only through their tail in the invariant mass distribution, thanks to the excellent \Ks{} mass resolution of about 4 \MeVM{} achieved by LHCb. A data-driven method to allow the training of the BDT in real data, avoiding the usage of the simulation, has been developed in this work of thesis. To avoid potential bias in the decisions taken in the analysis, the signal region ([492, 504] \MeVM{}) was not considered for the entire work of thesis. A preliminary estimation of the background expectation from a fit into the data mass sidebands after the rejection of the combinatorial background and the \Kspp{} tail is obtained and it is found to consist of a few events, as was the case in the 2011 result. This result is achieved thanks to the developed data-driven method, an improved muon identification algorithm, and the optimization of the muon identification requirement. From the results of this work it is possible to predict that, in case of no signal observation, a limit will be set that will likely overcome the current result by more than a factor 4, achieved thanks to the double integrated luminosity and, moreover, to the better trigger efficiency, measured in this thesis to be about 2.5\% (a factor of about 3 larger than the 2011 case). The result from this new analysis on the integrated luminosity of 2 $\mathrm{fb}^{-1}$ acquired by LHCb in the 2012, will be published by the LHCb Collaboration when the analysis will be completed and approved after a LHCb internal review. Only at that stage, events in the signal region will be counted and a result for the branching ratio will be given.