I'm not a fan of the sleeper myself - it really is not particularly nice, nothing like a hotel room and standards of cleanliness in my experience are, well, British Rail. OK if you're not that fussed about that too much and are happy to rinse it away with a shower at the other end (at a proper hotel) then fine but I know plenty of people I would not consider inflicting it upon.

Personally, I would try and go up the night before if at all possible (particularly if not attending the meeting is career limiting!) or get one of the first flights of the morning. LCY is absolutely fine *unless* the weather (fog usually) becomes an issue and with weather you will generally know a day or two ahead if it is going to cause you problems which gives you a chance to plan an alternate.