You do NOT have to figure this out before you file. Why try? When your assignment ends, file for benefits. Immediately. They'll tell you whether or not you qualify, and their information will be much more valid than what you get from talking to other people who don't know any more about it than you do.
File in the state where you live, where you wish to register for work. But regardless of where you file, you will be filing a KY claim.
What you say about WV unemployment being based on the last quarter is not correct, but there's no need to go into it. WV unemployment is not going to based on anything at all for you anyway, because you do not have enough WV wages to set up a WV claim. You will be drawing a KY claim, which was set up less than a year ago based on your previous work during the last 18 months.
What they are going to be looking for, in order to re-open your KY claim, which will be in place for a year since the time you filed it. (So if you filed it in June, 2011, and were denied, you'll be filing to re-open your KY claim now, since it has not been a year.)is called re-earnings. Re-earnings is an amount of wages from a covered employer in any state that you have to have made since your claim was denied, usually 10x the weekly benefit amount, or 5x the weekly benefit amount, before you can remove the denial and begin receiving benefits.
After the 15th, file at once to re-open your KY claim, using your final check stub from the place where your temporary job ended to show how much in gross wages you have earned working at this job. They will ask the right questions, you do not have to "argue your case" that you should get unemployment benefits.
The job ended through no fault of your own, which means you will likely be approved, providing you have enough re-earnings. Enough re-earnings plus a no fault separation means you'll likely be approved. If you are still drawing in May or June of next year, a year from the date this claim was signed, this one will end, and another claim would have to be set up, combining your KY wages and your WV wages, and you may or may not be eligible for a new claim, whether or not you have found a new job.
The employer in KY can ask the system in KY not to charge this claim to their account, which will have no effect on your claim right now. The claim has already been set up using wages paid by them. Your voluntary quit from them has already been ajudicated and your claim has been denied. However, if you check your denial decision letter, it will tell you that you have to work xx amount of time and make xxx amount of gross wages to remove this denial. It appears you may have done this. If so, you'll be approved.
Incidentally, if you are approved to draw KY unemployment again after this temporary work, the benefits will be paid from the general pool of benefits, but your KY employer's tax rates will likely be unaffected, since you did quit that job. Your employer in WV did pay a quarter, or possibly two, of wages for you, and then let you go. This period of employment will go into your whole unemployment quarter amounts and would be used later if necessary.
Long story short, don't try to figure it out yourself . Sign up for it and let the system figure it out for you. It doesn't cost anything to file for benefits, and you are much more likely to get good information if you file, rather than sitting back asking a lot of questions to somebody besides the system itself.