After a punishment, there is approximately a two week (of active playing) period of very strict enforcement. During that time, you have to be on your best behavior, or the IFS will decide that you didn't want to change your habits.
Once that is over, it takes 3 to 6 months of regular playing for your punishment level to decrease one level. This goes faster if there are fewer valid reports against you, and can take significantly longer if you're validly reported regularly.
The punishment tiers are:
Completely Clean
10 Game Chat Restriction
25 Game Chat Restriction
14 Day Ban
Permanent Ban
That was in reference to the punishment system (IFS), but it looks like you're asking more about the Honor system, so I'll see what I can explain.
After a 14 day ban your honor is set to level 0, and locked. After a period of time that is entirely dependent upon you putting in games where you are not validly reported, your honor will be unlocked.
Once honor is unlocked, you can start to accumulate honor again.
Honor level 0 through 2 takes a very long time to accomplish, but it can be done. Once you are level 1, you start receiving key fragments again. At level 2, key fragments drop at the regular rate.
So, if you want to avoid further punishment on this account, and recover honor ranking more quickly, I'd avoid a couple things that I know the IFS flags as negative: asking for reports, criticizing champion choice (time travel doesn't work yet, so talking about it during the game can't change anything), saying that someone is feeding or trolling (the IFS wants you to just report them, since they and everyone else already knows), and saying that you're muting and reporting someone (it's a threat against their account, and all threats are considered negative).