The best figures to use for the USA are the FBI UCR (http://www.fbi.gov/about-us/cjis/ucr/crime-in-the-u.s/2011/crime-in-the-u.s.-2011)
2011 figures are much lower (murder rates) and show the continual decline in murder/violent crime in the USA.
One point, comparing the rate of non-firearm murder versus UK non-firearm murder is instructive. Non-firearm murder in the USA during 2011 occured at about 1.5 per 100,000, but the question is whether those firearm murders would not be “converted” to non firearm murders.
Given that as many as 1/3rd of all murder in the US occurs in a “preexisting criminal relationship” (i.e. drug transaction, gang hit, etc) it is very likely the conversion rate to non-firearm murder would be significant. For instance, the rate of all murder in UK is 1.4 per 100,000 people, but in the USA the rate of murder-by-knife and murder-by-bare-hands and murder-by-blunt-object (2918 total) is 1.16 per 100,000 people. It is very reasonable to assume that the nearly 4100 murders where a criminal relationship exists would not convert to non-firearm murders given the frequency that Americans like to kill each other.
The incidence of murder-by-knife (1694 total in 2011) is nearly triple the number of murders by rifle or shotgun (the weapon categories to which “assault weapons” are assigned — i.e. AK47s and AR15 rifles — 679 total murders for both categories combined).