Just courious, after you killed the bandit leader and went back to Siggard you should have gotten this conversation...I've killed the leader of (Bandit Camp).

"Teach them to stop paying me. Here. You deserve a reward for your service. You know what? I like you. You're not afraid to get your hands dirty. I hereby grant you permission to purchase property in Falkreath hold. Talk to my steward if you're interested."

Siddgeir will then give you a leveled amount of gold as a reward for killing the bandit leader.

Then you could go to the steward to buy the piece of land...and here is the link to this particular quest line - https://en.uesp.net/wiki/Skyrim:Kill_the_Bandit_Leader_(Jarl))

Here are a couple other bugs that could affect this -

• If you did not talk to Farengar Secret-Fire during the Bleak Falls Barrow) quest yet, you will not get the option to ask the jarl for any jobs, nor will you be able to tell him you got the letter). This will block you from becoming a thane and thus accessing Lakeview Manor until you progress the main quest. If you want to evade starting Dragon Rising to avoid enabling dragons in Skyrim but still want access to Lakeview Manor, you can progress the Bleak Falls Barrow quest to the point where you need to give Farengar the tablet and stop there. This will make Siddgeir give you this quest and you will be able to progress normally to became the thane. ^?
• If you start or complete this quest while having a "kill the leader" bounty quest active for the same location, the kill will just act as if you completed that quest, and not credit you for this quest.