It doesn't look like the bed can be raised too much because you would be taking the soil over the brick/slab line, and therefore over the weep-holes above your foundation. Mulch will also be constantly spilling onto your sidewalk, too.
I would connect two six inch french drain box inlets with atrium grates on top, connect to 3" corrugated pipe, run under the sidewalk to a pop-up flush emitter in your yard. It's a really simple job for a landscape contractor in your area. You can get all the parts and advice on how to do it yourself at Ewing Irrigation supply.
Hope that helps,
Mitch.
_________________
Mitchell J McGowan
Landscape Contractor
Owner,
Dotdirt Organic Landscapes,
www.dotdirt.comGarland, TX