however from some years ago, when i learned the PIPE cloning approach, i'm not use any kit also but i'm just follow the PIPE cloning protocols as reported in the following paper (i attached the full text)
The Polymerase Incomplete Primer Extension (PIPE) Method Applied to High-Throughput Cloning and Site-Directed Mutagenesis | SpringerLink
it is quite simple, you need just to desing primers with 15bp overlapping regions which contain the mutation, perform a vector PCR with and high fidelity polimerase as KApa Hifi or CLone Amp and trasform the unpurified PCR in some specific E.coli strains, as MACH1 (Thermo) able to link it.
If you are interested i shared some more information about PIPE cloning in my sceintific blog, ProteoCool, where at the following link you can find some free accesibble and downlodable video about it:
I also like the Quickchange kit but 13 kb is really big for it. It is probably best to transfer the piece of DNA to be mutated into a smaller plasmid and then transfer it back to the vector you need it in. An additional advice: If possible design the mutation in a way that creates or destroys a restriction site. This makes it easier to discriminate between the different alleles.