There are a few teams in the first round that traded out their 2/3/4 round picks that might dance.
Oilers, Columbus, Devils ...
Probably a bit unlikely to move the Oilers (#14) or the Devils (#18/#20) to trade back to #35. If the Devils move a pick, I'd imagine it'd be for some immediate help.
The Kings have picks #35, #50, #58, #65, #82, #96, #111, #127, #158, #189. Some comparable trade ups in the last decade:
2010
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Islanders: #30 (Brock Nelson)
Blackhawks: #35 (Ludvig Rensfeldt), #58 (Kent Simpson)
2011
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Senators: #24 (Matt Puempel)
Red Wings: #35 (Tomas Jurco), #48 (Xavier Ouellet)
2014
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Islanders: #28 (Joshua Ho-Sang)
Lightning: #35 (Dominik Masin), #57 (Jonathan MacLeod)
2015
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Islanders: #28 (Anthony Beauvillier)
Lightning: #33 (Mitchell Stephens), #72 (Anthony Cirelli)
Blue Jackets: #29 (Gabriel Carlsson)
Maple Leafs: #34 (Travis Dermott), #68 (Martins Dzierkals)
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Ottawa has a boat load of picks as well, so the Kings might have some competition with moving up.