In my opinion, it needs to act quickly and decisively to try to hold on to some level of credibility.
1) There needs to be full disclosure, and the public needs to see the full findings of the investigation to see exactly how widespread the corruption has been. If it is to change public opinion, transparency is going to be key. The more that it tries to hide what has gone on, the less chance it has of recovering from this scandal.
2) Long-term, it needs to put in place a structure to ensure that the corruption that has taken place can never happen again.
3) If it is proven that the votes for the World Cups in Qatar and Russia were indeed the result of bribes, then FIFA has no choice but to organise a new vote otherwise everything about both these World Cups will be tainted and its own reputation damaged further.
4) There needs to be decisive action from within FIFA with all members of the organisation that have taken part in any criminal activity.
5) Sepp Blatter needs to leave immediately, and the next president needs to be an external appointment. Blatter cannot possibly remain there and be involved in the appointment of his successor. This is an opportunity for FIFA to show the world it is serious about reform. It needs to show that this is a completely new era for FIFA, and that cannot work if the old guard are still there influencing decisions. It needs a new president who can win confidence of the football associations and the wider football community.
6) Most people feel FIFA as an organisation is completely outdated and out of touch with what is going on in football. The football world has changed dramatically over the past 20 years, and it is apparent that FIFA hasn't. Its executive committee is mostly men, with an average age of about 64, and there have only ever been three women on the executive committee, with the first appointed in 2013. In my opinion it needs to appoint more women and its committee needs to be more reflective of the diversity of the football world.
7) FIFA is supposed to represent the sport around the world, yet seems totally disconnected to the footballing community: the players, clubs, officials, media and most importantly the fans. I think there needs to be more engagement between the organisation and real football fans.
8) Most importantly, it needs to begin repairing relationships with the media. Many media outlets over the years have made accusations of corruption but it seemed like FIFA felt it was untouchable, and was not answerable to anyone. It has found it isn't, and it needs to build these relationships, which again can only be done with full disclosure. Quick, clear and open communication from FIFA from now on is going to be key to changing perceptions.
With the FBI investigation sparking new revelations daily, it seems there is still much more to uncover, and if FIFA wants to have any chance of surviving with some sort of credibility then it needs to act quickly. Changing the reputation and public perception of FIFA is going to be a long and slow process.