Thanks a lot. I think I've got all that so I'd better clean up the comments. ^^v I wasn't aware that that use of ように can sometimes omit the 'に'.
本当に申し訳ないのですが、私はあなたの傘をなくしたらしいのです。 This sentence sounds awkward. I would say "なくしたようです" instead of "なくしたらしいのです."
Some people would say "なくしたらしい（のです）。".
Could someone tell me what's the difference between "Demo, Keredomo" and "Ga"?
as far as i know, ga is more like a "yet" type of but. demo means something closer to "however". And keredemo is closer to "nevertheless".
oh, i should add that ga typically contrasts between two differing sentences. Like that one above, 毎日一生懸命に英語を勉強したが、あまり身につかなかった。 I studied English very hard every day, but I did not learn a lot. a demo sentence would be この本はすてきでも買えないだからすぎる高いですよ． a keredemo sentece implies that you've done something despite something else. It would be この本は高いけれでも買いました． Feel free to correct any of my mistakes but these are the rules as i learned them.