Most methods work fine. I would suggest that you use whatever method your laboratory is familiar with and has used in the past, so that there are people in the lab that can help you troubleshoot if it doesn't work. Or a neighboring lab. If this is a new technique for your lab, then I would probably suggest just getting a kit where you know the reagents are all working well.
No need for a kit. Google the basics ("Quick change protocol") and get a non-proofreading polymerase, DpnI to digest methylated DNA templates and 2 primers carrying the desired mutation (Edit: and a ligase, eg from T4). All dog cheap stuff. 90% of the price of such a kit you're paying for the manual.
Then grow some clones, check the right size of the insert and sequence them.
Design your primers and then Quick Change. You can either use a kit https://www.agilent.com/cs/library/usermanuals/Public/200523.pdf or go with Wolfgang`s suggestion. It`s super easy!
I recommend to use the entire kit as mentionned above (Quick Change), because it also contains specific competent cells that greatly optimize the reaction. I always obtained 100% success with this kit (and my colleagues in the department too). Just an advice, use SOC medium (it works well).
Patricia, a kit makes good sense if you're doing this as a one-time shot and you'll never do mutagenesis again. In a "normal" molecular biology lab however, the only thing that might be missing is DpnI. Which is a good investment by itself if you want to make sure your PCR template carryovers get efficiently destroyed before you find them in your clones.