First of all, I understand your grief. What parent could not? I have a daughter just a few years older than yours. I also understand your desire to have something good to come out of your daughter's life and death. I hope to help you in that regard.
You have an opportunity to do things differently than others who have faced similar tragedies. You can start by reaching out to the gun rights community. You may be shocked to find out that we actually want our gun laws to work. We too want to prevent as many crazy people as possible from getting their hands on firearms. We can also tell you why our current laws are failing. One more thing: The political reality is that nothing will be passed at a federal level without the support of gun rights activists. If you really want to get something done, you cannot alienate us.
Additionally, we have to work within the limitations placed upon all constitutional rights. A law that is simply aimed at reducing the number of guns in the country would not only accomplish little - it would be as unconstitutional as a law limiting free speech or regulating religion.
The good news is that gun rights advocates want these problems fixed. We want the background check system to work. In fact, we have been working to fix some of them, with little to no support from gun control advocates, and zero press coverage. There is much that can be done with zero resistance from the gun rights community.
The problem is not primarily a lack of "gun laws" - it is a failure in the support system that is needed to make them work. This is why we react so strongly against calls for more laws. The fact is that, while we would be open to minor changes, we have all the gun laws we need. The changes are needed elsewhere - as I will detail below.
Here are the problems that are causing our background check system to fail:
Massive under reporting of disqualified persons.
It is estimated that at least one third of people who have been prohibited by law from owning firearms have not been reported to the FBI for inclusion in the background check database. The murderer at Virginia Tech should not have been able to buy a firearm due to his mental health history. After this incident the NRA called for better reporting and supported legislation to that effect - even though this caused concern among many members. Sadly, even the passage of this law has failed to solve this problem - as the recent racist church shooting sadly proved. It may surprise you that the gun industry has continued to work to improve reporting. See http://www.fixnics.org
Virtually zero enforcement of laws against trying to buy a gun illegally.
Sadly, it has become common knowledge that prohibited persons have at least a one in three chance of buying a firearm through legal channels. So criminals and the mentally ill frequently try to buy guns at licensed dealers. Although they stand a good chance of being rejected, they have less than a 1 in 300 chance of being prosecuted. Although tens of thousands of people are rejected each year, prosecutions run less than 20 per year. The Obama administration has actually said that such prosecutions are "a waste of resources". In fact, the feds do not even notify local authorities - who frequently could bring state charges or violate the criminal's parole or probation. Until there is enforcement of this law - violation of which is a felony - then criminals and the mentally ill will continue to try get guns from dealers.
Failure of the mental health system to identify and treat dangerous people.
The Supreme Court has made it clear that severely mentally ill people can be banned from owning or possessing firearms. However, common sense tells us that before someone can be banned from owning firearms they must be identified. Sadly, many of the murderers in the recent incidents had displayed bizarre and disturbing behavior. Those who perpetrated the Giffords shooting the Isla Vista stabbing/shooting and the Navy Yard murders, had many police contacts that SHOULD have resulted in a mandatory mental health evaluation, but did not. Police are the eyes and ears of the mental health system - if they leave mentally ill people on the street, the system fails. If they send them in for evaluation, with a good description of why they are doing so, then there is a good chance that action will be taken that would result in a firearms ban and reporting to the FBI background check database. If this never happens, they will continue to be able to pass a background check.
Depriving someone of a constitutional right requires due process.
The reality is that firearms ownership is a constitutional right. In criminal cases, there is no problem - because due process is well established. In mental health cases, the federal disability also requires due process - specifically a mental health commitment by a court. In order to be legal, any additions to the law would also have to provide for due process. Furthermore, a person cannot be required to prove their sanity in order to exercise a constitutional right. The burden of proof rests upon the government. Additionally, I would ask you to consider this: If someone cannot be trusted with a firearm due to mental defect, should they be left on the street? I think not - because that leaves them a lot of options for harming people. In addition to illegally buying a gun on the black market, they could build a bomb, start a fire or run people down with a car. If they cannot be trusted with a gun, they need treatment - and we must insure that they get it. After this, I have no problem with a law that requires them to prove their sanity before being allowed to possess a firearm.
Once the background check system is truly fixed, it will be much easier to expand it to private sales.
At this point the vast majority of lawful sales, including those at gun shows, go though the background check system (the 40% figure frequently used is bogus - it is based on a phone survey that covered two full years before the background check was in place). As outlined above, it is a deeply flawed system that most gun rights organizations want fixed. On the other hand, most gun control organizations completely ignore the flaws in the system, and instead want it expanded to private sales before it is fixed. This is one of the biggest reasons why so many gun owners are opposed to expanding background checks - we don't trust the motives of groups that don't care that the current system doesn't work. Frankly, it seems to many of us that some of the gun control groups WANT the current system to continue to fail - because if it were to work, their would be less chance of passing their draconian restrictions. Unlike yourself, many of these people want an outright repeal of the 2nd Amendment.
On the other hand, if we fix the current system and it begins to actually work. it will be seen that the motive is solely to do all we can to keep guns out of the wrong hands - and the greater the support will be among gun rights activists will be for expanding such checks to private sales.
You will remain in my prayers and in the prayers of many other Americans. It is my hope that you will realize your stated goal of preventing as many mentally ill people as possible from harming others. If you reach out in good faith to the gun rights community, you will find that we share that very same goal.