Well before you start building anything new for planting nearby, its probably best to dig around and expose the major woody roots on the elm - I assume it was originally only one elm, and its now got shoots coming off the main limbs below ground. If that's the case, then reveal as much woody root as you can, drill into it to make pits about quarter to half inch deep, or make deep cuts in it, use a tree stump killer by applying it to the cuts/holes you've made (I'd say SBK, called a brushwood killer in the UK, but don't know what types are available where you are), cover with something and leave to die back. You can push the soil back over the top if the areas you've treated are well covered. If you're careful when you use whatever tree stumpkiller you choose, not getting it anywhere other than on the offending roots, it shouldn't preclude making preparations for your keyhole garden. The stumpkiller won't make the roots disappear any quicker, but it will stop regrowth - decomposition of the dead roots will then proceed over time.
If you've got shoots coming off the stump of the tree above ground, then its worth treating that too by drilling into it, round the outside edge, don't bother to treat the heartwood, and applying stump killer. In fact, if that's the only place where new shoots arise, you don't need to worry so much about poisoning all the roots.