~2 Hours after a large insert SQL Server gets "I/O requests taking longer than 15 seconds to complete"
BOL refers to 833 event ID as :


  This problem can be caused system performance issues, hardware errors, firmware errors, device driver problems, or filter driver intervention in the IO process.


What it means is that 


SQL server is doing more I/O's that what the disk subsystem can handle.
There could be some rogue process running on the system that is saturating the disks with all the I/O requests.


Below is the approach I would follow:

Troubleshoot using Wait Statistics - DMV's

WITH [Waits] AS
    (SELECT
        [wait_type],
        [wait_time_ms] / 1000.0 AS [WaitS],
        ([wait_time_ms] - [signal_wait_time_ms]) / 1000.0 AS [ResourceS],
        [signal_wait_time_ms] / 1000.0 AS [SignalS],
        [waiting_tasks_count] AS [WaitCount],
        100.0 * [wait_time_ms] / SUM ([wait_time_ms]) OVER() AS [Percentage],
        ROW_NUMBER() OVER(ORDER BY [wait_time_ms] DESC) AS [RowNum]
    FROM sys.dm_os_wait_stats
    WHERE [wait_type] NOT IN (
        N'CLR_SEMAPHORE',    N'LAZYWRITER_SLEEP',
        N'RESOURCE_QUEUE',   N'SQLTRACE_BUFFER_FLUSH',
        N'SLEEP_TASK',       N'SLEEP_SYSTEMTASK',
        N'WAITFOR',          N'HADR_FILESTREAM_IOMGR_IOCOMPLETION',
        N'CHECKPOINT_QUEUE', N'REQUEST_FOR_DEADLOCK_SEARCH',
        N'XE_TIMER_EVENT',   N'XE_DISPATCHER_JOIN',
        N'LOGMGR_QUEUE',     N'FT_IFTS_SCHEDULER_IDLE_WAIT',
        N'BROKER_TASK_STOP', N'CLR_MANUAL_EVENT',
        N'CLR_AUTO_EVENT',   N'DISPATCHER_QUEUE_SEMAPHORE',
        N'TRACEWRITE',       N'XE_DISPATCHER_WAIT',
        N'BROKER_TO_FLUSH',  N'BROKER_EVENTHANDLER',
        N'FT_IFTSHC_MUTEX',  N'SQLTRACE_INCREMENTAL_FLUSH_SLEEP',
        N'DIRTY_PAGE_POLL',  N'SP_SERVER_DIAGNOSTICS_SLEEP')
    )
SELECT
    [W1].[wait_type] AS [WaitType],
    CAST ([W1].[WaitS] AS DECIMAL(14, 2)) AS [Wait_S],
    CAST ([W1].[ResourceS] AS DECIMAL(14, 2)) AS [Resource_S],
    CAST ([W1].[SignalS] AS DECIMAL(14, 2)) AS [Signal_S],
    [W1].[WaitCount] AS [WaitCount],
    CAST ([W1].[Percentage] AS DECIMAL(4, 2)) AS [Percentage],
    CAST (([W1].[WaitS] / [W1].[WaitCount]) AS DECIMAL (14, 4)) AS [AvgWait_S],
    CAST (([W1].[ResourceS] / [W1].[WaitCount]) AS DECIMAL (14, 4)) AS [AvgRes_S],
    CAST (([W1].[SignalS] / [W1].[WaitCount]) AS DECIMAL (14, 4)) AS [AvgSig_S]
FROM [Waits] AS [W1]
INNER JOIN [Waits] AS [W2]
    ON [W2].[RowNum] &lt;= [W1].[RowNum]
GROUP BY [W1].[RowNum], [W1].[wait_type], [W1].[WaitS],
    [W1].[ResourceS], [W1].[SignalS], [W1].[WaitCount], [W1].[Percentage]
HAVING SUM ([W2].[Percentage]) - [W1].[Percentage] &lt; 95; -- percentage threshold
GO


Look at the pending I/O's 

SELECT database_id
    ,file_id
    ,io_stall
    ,io_pending_ms_ticks
    ,scheduler_address
FROM sys.dm_io_virtual_file_stats(NULL, NULL) t1
    ,sys.dm_io_pending_io_requests AS t2
WHERE t1.file_handle = t2.io_handle


Below are PERFMON counters that will help you:


Avg. Disk sec/Transfer => Time taken to perform the I/O operation
Disk Bytes /sec  => Total reads and writes to disk per second in bytes.
Process:IO Data Bytes/Sec => Total reads and writes to disk per second in bytes by each process.
Buffer Manager: Page Read/sec + Page Writes/sec =>Total reads and writes to disk per second in bytes by SQL Server process.


Also, sometimes such errors can also occur when a different Power Management scheme is selected for the server. So better check your Windows Power Plan setting, and make sure you are using the High Performance Power Plan. 


  All three stored procedures had flaws which made them insert not a few hundred records but over 500.000 each time they ran. (These are now fixed so the issue doesn't present itself but I feel there is an underlying problem here).


If you are inserting a huge amount of data, then you need to refer to The Data Loading Performance Guide to cater your server for such operations e.g. enabling trace flag 610.

Note: Others have mentioned that Auto Update Stats might be causing it, which is also worth looking at, but I would primarily concentrate on finding out rouge process that is causing lots of Disk I/Os and checking the storage subsystem to see if all the divers, etc are up-to-date and is configured correctly.

Good reading links :


I/O requests taking longer than 15 seconds to complete on file
Troubleshooting SQL Server I/O requests taking longer than 15 seconds - I/O stalls &amp; Disk latency
Event ID 833: I/O requests taking longer than 15 seconds