Yup, that's the Taylor K-2005 that Leslie's rebrands. So you can just get the FAS/DPD chlorine kit from TFTestkits and you'll be good to go.
In the meantime, you can go ahead and look at the Pool School and use the test kit you have. The only difference will be that for the current chlorine test you have, you compare color against a comparator and it will be limited in both range and accuracy. It's better than nothing, though.
If you follow the instructions that came with your kit, you'll read chlorine and pH as numbers directly off the comparator tubes so the chlorine reading will be in ppm (i.e. you don't have to do any math to get it). For other tests, each drop will be a certain ppm and it will tell you in the test. For the TA and CH tests, it's 10 ppm per drop when using a 25 ml sample size or it's 25 ppm per drop when using a 10 ml sample size. The CYA test has markings in a tube that are already in ppm that you read directly.