Database which was set up for logshipping is dropped I dropped an old unused database. Then I found it was set up to be the source database of a logshipping setups. And there are a lot failing jobs in our job running status report now.
How to remove the logshipping settings after the database is already removed?
In order to fix alert issues use database: msdb and

First of all find out details on Primary Server by executing following..

PRIMARY SEVERE 

SELECT * FROM dbo.log_shipping_monitor_primary
where primary_database like 'CHANGE_DB_NAME%'


SELECT * FROM dbo.log_shipping_primary_databases
where primary_database like 'DB_NAME_HERE%'

SELECT * FROM dbo.log_shipping_primary_secondaries
where secondary_database like 'DB_NAME_HERE%'


Once verifies DB names execute 

delete FROM dbo.log_shipping_monitor_primary
where primary_database like 'DB_NAME_HERE%'


delete FROM dbo.log_shipping_primary_databases
where primary_database like 'DB_NAME_HERE%'

delete FROM dbo.log_shipping_primary_secondaries
where secondary_database like 'DB_NAME_HERE%'


Follow same steps on SECONDARY SERVER 

SECONDARY

SELECT * FROM dbo.log_shipping_monitor_secondary
where secondary_database like 'DB_NAME_HERE%'


SELECT * FROM dbo.log_shipping_secondary_databases
where secondary_database like 'DB_NAME_HERE%'

SELECT * FROM [dbo].[log_shipping_monitor_primary]
where primary_database like 'DB_NAME_HERE%'


and once confirmed delete details from table by executing following....

delete FROM dbo.log_shipping_monitor_secondary
where secondary_database like 'DB_NAME_HERE%'


delete FROM dbo.log_shipping_secondary_databases
where secondary_database like 'DB_NAME_HERE%'

delete FROM [dbo].[log_shipping_monitor_primary]
where primary_database like 'DB_NAME_HERE%'


Hope this will help you to clear your unnecessary notifications from msdb database. It happens when you do not clear log-shipping clearly or in your case database dropped.