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I've hiked a lot in Swedish Lapland, and I used to live there for nearly six years. The mosquitoes can indeed be very bad in summer, in particular in the forest.
Mosquitoes wake up between mid and late June and are worst at the beginning. For a very early season hike, you might have none at all if you go just before they wake up, or you might get the worst mosquito experience you can imagine¹. Hiking Kungsleden before midsummer is not realistic in a normal year, as there will be too much snow. Some sections require seasonal bridges or seasonal boats. Hikers have gotten stuck due to hiking out of season, unaware the boats wouldn't be in place, and having not enough food to hike all the way back or around. Don't be that person. Don't try to hike Kungsleden before the mosquitoes come; it's not usually feasible.
July is bad. July in the forest can be very bad. Much of the forest is swampland. Avoid to be anywhere below the treeline in Swedish Lapland in July, unless you are a natural predator of mosquitoes. Above the treeline is better as there's usually some wind to blow the mosquitoes away.
August is less bad than July. From about mid August, the mosquito density in the forest becomes quite bearable. August is quite good for a hike, although I'd still bring insect repellent.
Late August to mid September is the best season for hiking for several reasons. The landscape is spectacularly colourful. There are no more mosquitoes — the night frost kills them. There is likely aurora borealis (northern lights) at night. The nights are dark allowing you to sleep. There are less crowds. Crowds? Yes, Kungsleden gets crowded in summer — if you want to avoid crowds, consider Nordkalottruta (regardless of season) instead. If you hike Nordkalottruta or Grensesømmen in September, you will meet few other hikers (perhaps none at all), but even Kungsleden starts to be OK by then, in particular in the more quiet sections you may only meet several groups per day.
Around mid September, the trees lose their leaves. The landscape starts to lose its spectacular colours. Snowfall is common, and unlike August or early September, snow may not melt within hours. From around early to mid October, snowfall will not melt at all until following June. On the plus side, if you hike late September, you will have the mountain all for yourself, even on Kungsleden (but the same problem as before midsummer applies, as seasonal bridges and boats may have been removed; I don't think exact removal dates are publicly announced, but for boats certainly before the lakes start to freeze over around mid-October; the only way to be 100% sure the boats are in place is when the huts are open, check the STF website for dates).
One of the most spectacular hikes I've done was four days in October on Nordkalottruta between Vuodnabahta and Sørskjomen, but I lived in Kiruna at the time so I knew the conditions. We were the last visitors for the next 5–6 months there.
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¹Not Kungsleden, but if you're feeling masochist, spend midsummer or early to mid July in the valley of the upper Könkämä river — you will forever remember any Lapland mosquito experience in "not as bad as" terms. As I was cycling along the road (almost without breaks, as there was only one slightly windy spot for 100 km) I saw people fishing with head-to-toe mosquito nets, and despite midnight sun it was quite dark in my (thankfully mosquito-proof) tent in the morning, as it was entirely covered in mosquitoes.
