Hello new Lee player! I'm a Lee main, and I've got some helpful advice. What you're experiencing is COMPLETELY normal. In fact, Lee's late game is what allows Lee to be so strong. First off, your early game is amazing with ganks galore and a ton of early base damage. As the game continues, it is your job to control the map. This means taking out strong/fed opponents and snowballing lanes to the best of your ability. However, when late game comes you have to take on a new role. You have a choice on your role depending on your build;
Tank: Dive into team fights and deliver the enemy carry to your team's doormat
Damage: Versatile Assassin that leaps into the enemy carry, eliminates the threat in .5 seconds and escapes without even breaking a sweat.
Now whatever suits your playstyle and ability you can do, however I prefer to base it on how well the team is doing and the composition and synergy of the team itself. Be that as it may, if the enemy has a 11/2 Zed dicing up my homies then I'll go tank.
Typically for damage items, you're gonna want to go with these:
High Priority:

These items are crazy good on Lee. With Sheen procs and AOE damage to feed a small african villiage for an entire year, you should have no problem melting down those pesky plumbers
Considerable:

These guys are pretty nifty. The HP from Frozen Mallet and the MR in the Maw of Malmordius helps out a ton when dealing with APCs. Now if there was enough space in the high priority spot I would have added Black Cleaver, because it works so well with him. The Armor shred, the HP, it's like a Christmas
Honerable Mentions:

These guys are decent on Lee, so don't completely overlook them, but they won't be as effective as the latter
Now for tank, you're gonna see a lot of familiar faces:
High Priority:

These items typically help with sticking potential, bonus lifesteal/spellvamp on W (from Spirit Visage) and DPS magic damage. Very helpful and effective on Lee
Considerable:

This set consists of very over-the-top counter items for emergencies. Thornmail should be built whenever you face an auto attack-based carry such as Tryndamere, Master Yi, Udyr, Vayne, etc. Warmogs and Spirit Visage are a great combo that allow you to regen MASSIVE amounts of health in a shor period of time.
Tank doesn't have any honorable mentions, because no other tank items synergize with Lee as well as these
Lastly, here are some common-every-game things to keep in mind:

This is your best friend. Whether I'm tank Lee, or Damage Lee, I always pick this up. This allows you to not only stick to opponents, but gain armor penetration and AD. Meaning that even if you are tanky, you can still do lots of damage.

This is an absolute must-have item on Lee. Not only does it provide succulent vision of the map, but placing it over a wall and casting W on it can get you out of a lot of sticky situations that anybody else would have died in.
And as a final tip, always remember to take advantage of your energy regen passive in teamfights so your spell costs don't hinder you as much. Have fun with my favorite champ and see you out on the Rift!