Recently I've been trying to make something like the refreshing Jasmine Lime Cooler by brewing a big pot of jasmine tea, steeping for an hour or two and keeping that in the fridge to add a splash of lime cordial to (Bottle Green rather than Roses, as it has a lot more lime and a lot less sugar). I'm not sure if it needs fresh lime or a bigger slug of cordial to approximate US limeade - or margarita mix ;-) - as it's not quite as tangy and limy as Peet's.
Usually I wouldn't brew jasmine more than a couple of minutes; for drinking hot, long brewing brings out far too much tannin that overwhelms the jasmine. For drinking cold I think it needs the longer brewing to get strong enough to retain flavour in a cold drink; I often brew a second pot with the leaves after I've put the first pot in a jug and it's too light in flavour to stand up to the lime as much, but very refreshing.
Obligatory nugget of incorrect common knowledge ;-) Brits should be called lemonies rather than limeys as Cook used lemons to avoid scurvy in his crew; limes don't have enough vitamin C to help.