No - I don't think you could go closer to the wind as that is down to to the sail settings.

The Centre/dagger board stops the boat from going sideways. Changing the sail setting/pattern would affect sailing closer to the wind.

You could always try this with your Mirror. Ask someone to make a daggerboard double the length your's is, see how your boat performs!

All lifting surfaces; i.e. wings, keels, c'boards, rudders etc all suffer from losses of lift.  These include tip losses where the low pressure flow jumps over the end to join the high pressure flow (this is why commercial aircraft have little winglets on the end of each wing), venting when you have a rudder or c'board/keel operating close to the surface of the water and of course turbulent flow due to stall or surface finish. 

Now if you consider the aspect ratio of a keel or centreboard, that is the span (depth in our case) compared with its breath then a keel with a small length but a long root chord (the length against the hull) will suffer these losses against a greater proportion of its span and therefore will be inefficient.  A keel/c'board with a long span and short root chord with suffer these losses against a small proportion of its span hence being efficient.

Consider a jet plane verses a glinder.  A jet plane has dirty great big engines which can provide the speed to use smaller (i.e lower aspect ratio) less efficent wings.  A glider has no engine so it has long, high aspect ratio wings to be as efficeint as possible.

So to answer your question a c'board with a short span but long root chord would be easy to steer in a straight line, be hard to tack and would slip sideways!!!

Long spans, short chords tach easily, don't slip to leeward as much but stall easily so its a compromise between reliability and performance.

Hope this helps