I know Bronze and Silver are two different worlds entirely, but when I was stuck in a similar situation in Bronze 5 (people feeding way too hard, games being determined almost completely by which of the other 9 players feeds hardest) but then I did something interesting... I played a champion in a way that people weren't prepared for. I mean, don't do something stupid, like full-crit Lux top or something, but here's what I mean.
I had played Sion top and in the jungle for a long time. I knew his wind up times on his Q, the range and mechanics on his E, and the delay on his W. Heck, I also was somewhat decent wall drifting with his ultimate if I could get the angle right. But, I simply wasn't getting progress on either of those roles. Jungling in anything other than Gold is a HUGE risk. Unless you are the God of playing someone like Yi, Jax, or Pantheon or something, you probably aren't going to carry consistently.
But, you know what I noticed? Sion is actually a really strong support champion. He has great base damages, so his poke can be devastating. I also knew that players in bronze don't ward their bottom lane bushes, or, if they do, they don't consider the possibility of Sion being inside. Also armed with the knowledge that you can't see Sion's Q if you don't have vision of Sion, I would very easily catch people out.
Now, am I saying that Support Sion is the answer to all of you problems? Absolutely not. My win rate dropped from 95% to about 55% to 60% as I climbed up to Bronze 2. But, before that point, I was doing horribly, because I was playing champions that require your team to at least hold it together without you having to constantly babysit one or more other lanes. I was playing my current main, Veigar, who really can't stand lanes that lose the game for him early. I had some success, but my win rate was consistently below 40%. At the time, I thought that getting autofilled support meant that you might as well flip a coin to determine if you're going to win or not, because supporting with Lux or Lulu just wasn't cutting it for me.
What I found out was that I needed a different champion for the situation that I was in. Also, don't be afraid to try new champions that aren't played often, unless they have some kind of garish win rate like 40%. Logically speaking, it would probably be best to prey on players' weaknesses at that level. From what I gather, the consensus is that players at the Silver level have a somewhat rounded experience - they've probably played each role at least once. Often times, Silver players may have decent CS, and generally have very high mechanical skill with one or two champions, skewing towards simple champions like Annie, Caitlyn, Pantheon, etc. And, finally, Silver players don't ward as often as they should. If they do, then they don't check the mini map as often as they should.
So, considering you seem to be well-versed in the Jungle, I would recommend a champion that excels in ganking, and being unseen. Shaco is good, because his clearing is so safe that you can often counter-jungle the enemy very easily, and his ganks are easy to pull off, due to his ability to go into stealth. If his ultimate is giving you issues (because, trust me, I would never try playing such a complicated champion) Kha'zix would probably be a good alternative. Twitch is also good as a jungler, but requires a lot more kiting of jungle camps to be effective, and generally isn't as consistent as Shaco or Kha'zix. Rengar is also good, because, when played properly, he can severely punish people who like to face-check bushes, or run past bushes with no fear.
If you'd like an even safer bet, Zac is another great ganking champion. He can often drop in over walls, or sling-shot in from huge distances away. He also has enough crowd control that would facilitate allies to get kills. Sejuani is also good, because her ultimate is a godlike source of CC, has a huge area of effect, and she also has a charge that can go over walls, further extending her range. Vi is another good pick, because her Vault Breaker goes over walls and will cc, she usually builds bruiser so she's kinda tanky but outputs some serious damage, and her ultimate is fantastic for locking people down. She also reduces armor, so she's great for getting AD-based champions snowballing hard, especially if they're behind.
Now, I say this because that's what's been said to me. I am an AWFUL jungler, but the logic seems sound on these champions.
Also, according to your OP.gg, you tend to hang on to your warding trinket. As a jungler, it may be more effective to buy a sweeping lens, to make sure that your ganks can take place without detection. It may be a little different if you continue to play Kayn, if you decide to buy duskblade. But, considering in your past several games as Kayn, you haven't bought duskblade, it's probably not part of your core build. But, it is good to see that you're placing a good number of control wards down - it's something that not enough people do at lower levels.
But, then again, I'm bronze, and also not taking my own advice, so take from this what you will.