What limits our healing speed?
The process of wound healing is pretty complicated and is much more than only cell division. The woundhealing process starts almost immediately in the moment of the injury and depending on the injury can take quite a long time. Have a look at the figure below (from the second reference), which shows the phases:



Basically this can roughly be diveded into four phases:


Hemostasis/Coagulation:


vascular constriction (blood vessel contraction)
platelet aggregation, degranulation, and fibrinformation (formation of a thrombus and closure of the wound)

Inflammation:


neutrophil infiltration
monocyte infiltration and differentiation into different kinds of macrophages
lymphocyte infiltration

Proliferation:


re-epithelialization
angiogenesis (formation of new blood vessels)
collagen synthesis (formation of new connective tissue)
formation of the extracellular matrix

Remodeling:


collagen remodeling
vascular maturation and regression



This process takes some time and can be prolonged when a wound is infected.  So I don't think there is much room to speed this up, as you need to go through all these phases. This process is also the result of extensive evolutionary processes; otherwise, such complex cooperation between different cells is not possible. For more details on the process itself, have a look at the references.



References


Factors affecting wound healing.
Cellular aspects of wound healing.