There are a number of limitations on executive power. There is a far greater separation of powers in US government than in the UK parliamentary system, largely due to its artificial nature (with parliament being sporadic in growth). For instance, the executive is not drawn from the majority party of the elected legislature, meaning that the head of government may lead a party which is not in power in either chamber (at the moment, the House of Representatives has a Republican majority). The Senate also has the power to propose and indefinitely block legislation, which are not really features of the House of Lords. The whip system is largely absent, though there has been something similar imposed within the Republican party in the Senate. Finally, not immediately relevant, the judiciary has far greater powers than in the UK, which it acceded to in the case of Marbury v. Madison (it has an entrenched constitution to refer to, after all). To an extent, the UK emulated the US in separating the judiciary from the legislature. These features can have ugly consequences for something as apparently simple as acknowledging the implications of a budget, but it prevents hasty responses such as the one above.
Now onto the options Obama has available: there are varying degrees of executive actions, with the highest authority being given to executive orders. However, the scope of application and capacity to implement these is limited severely in comparison to the ability of the legislature. Extra-legal means are largely impossible, given both the particular nature of the populace in question (no vanguard to accomplish such a coup, likely to agitate counter-revolution) and the immediate impeachment likely to entail from the legislature or the challenges to such an action by the Supreme Court. From a practical perspective, this is a good thing. While shootings are tragic, they constitute far fewer American life-hours lost than due to heart disease and cancer. An executive action to promote exercise and nutrition would probably be more effective at stemming losses of life. Humans are hard-wired to think in terms of specifics though, research conducted by both Dunbar and Harris has demonstrated as much. See also the spotlight fallacy.