You were close, but the full condition should be in a single pair of square brackets.
DECLARE @tbl TABLE(id int NOT NULL PRIMARY KEY,
xml xml)
INSERT @tbl(id, xml)
VALUES(1, '<Jobs>
<Job JobName="Formulatrix Database Index Rebuilding.Subplan_1" JobEnabled="true" ScheduleName="Formulatrix Database Index Rebuilding.Subplan_1" ScheduleFrequency="Weekly" ScheduleInterval="Sat " ScheduleTime="01:00:00" ScheduleTimeDescription="Starting at 01:00:00" NextRunTime="2023-05-13 01:00:00" LastRunStatus="Successful" />
<Job JobName="Integrity_check_DBCC" JobEnabled="true" ScheduleName="monthly_integrity_check_schedule_wave2" ScheduleFrequency="Every 1 Months" ScheduleInterval="First Sat" ScheduleTime="20:00:00" ScheduleTimeDescription="Starting at 20:00:00" NextRunTime="2023-06-03 20:00:00" LastRunStatus="Successful" />
<Job JobName="syspolicy_purge_history" JobEnabled="true" ScheduleName="syspolicy_purge_history_schedule" ScheduleFrequency="Daily" ScheduleInterval="Every Day" ScheduleTime="02:00:00" ScheduleTimeDescription="Starting at 02:00:00" NextRunTime="2023-05-09 02:00:00" LastRunStatus="Successful" />
</Jobs>')
SELECT * FROM @tbl
SELECT xml.query('/Jobs/Job[contains(@JobName,"Integrity_check_DBCC")]')
FROM @tbl
WHERE (xml.exist('/Jobs/Job[contains(@JobName,"Integrity_check_DBCC") and @LastRunStatus="Failed"]') = 1);