well, as already stated the ak2i is a ds-mode flash kit and can not run in the dsi-mode. the ievo's dsi-mode homebrew runs very well. so far we have cquake, a quake 2 project in beta, sandbox engine, the gba emu, and alekmaul's atari emus which were coded specifically for the dsi-mode.
with cquake you can expect a higher frame rate and mp3 playback, with the atari emus you can expect games which run slow on the ds to run at full speed (or very close to it) on the dsi. quake 2 is in beta but its already running much faster and loads much quicker than the slot-2 ram expansion solution. sandbox engine is in development and i don't know much about its specifics to comment on it. the gba emu i haven't actually tried but some users have said that it works pretty well with gba homebrew.
many developers have recompiled their projects for the dsi-mode but they have not done any recoding to really make use of the faster cpu and extra ram. so while some things will play "faster" in dsi-mode (snemulds) it is often just the ds-mode code being run in the faster cpu.
if you can not get access to a dsiwarehax dsi then the only alternative is the ievo. so if you want to play around with dsi-mode homebrew, try to find one a bit cheaper because it seems that cyclops will never iron out the compatibility layer issues and they haven't updated it for awhile. you would be throwing money away if you picked one up with the intention of having high homebrew/rom compatibility and a dsi-mode flash kit.
-another world