Add a message handler for it to your main form, private section:

 procedure WMSpoolerstatus( var msg: TWMSpoolerstatus );
   message WM_SPOOLERSTATUS;
  
Hit Shift-Ctrl-C to make the IDE add a implementation for the method.
Add an inherited statement. msg.JobStatus will be 0 if the spooler thinks
it is healthy and < 0 if it has an error. From the list of error codes
found in Windows.PAS (above PR_JOBSTATUS) a missing network printer may not
be seen as an error, so look at msg.JobsLeft. If this never decreases in
the messages you see the spooler cannot get rid of the jobs. Of course it is
hard to tell when enough time has passed so you might become suspicious.

You can try to use the WinSpool.EnumJobs API to query the spooler directly.
Example below. See the diverse JOB_STATUS* codes given in the help topic for
JOB_INFO_1 in win32.hlp. Note that the Status field is a bitset, it can contain
more than one of the status codes. Use expressions like

 if (Status and JOB_STATUS_OFFLINE) <> 0 Then
   ... printer offline

uses Winspool, Printers;

function GetCurrentPrinterHandle: THandle;
var
  Device, Driver, Port : array[0..255] of char;
  hDeviceMode: THandle;
begin
  Printer.GetPrinter(Device, Driver, Port, hDeviceMode);
  if not OpenPrinter(@Device, Result, nil) then
    RaiseLastWin32Error;
end;

Function SavePChar( p: PChar ): PChar;
const error: PChar = 'Nil';
begin
  if not assigned( p ) then
    result := error
  else
    result := p;
end;         

procedure TForm1.Button2Click(Sender: TObject);
type
  TJobs = Array [0..1000] of JOB_INFO_1;
  PJobs = ^TJobs;
var
  hPrinter : THandle;
  bytesNeeded, numJobs, i: Cardinal;
  pJ: PJobs;
begin
  hPrinter:= GetCurrentPrinterHandle;
  try
    EnumJobs( hPrinter, 0, 1000, 1, Nil, 0, bytesNeeded,
              numJobs );
    pJ := AllocMem( bytesNeeded );
    If not EnumJobs( hPrinter, 0, 1000, 1, pJ, bytesNeeded,
                     bytesNeeded, numJobs )
    Then
      RaiseLastWin32Error;

    memo1.clear;
    if numJobs = 0 Then
      memo1.lines.add('No jobs in queue')
    else
      For i:= 0 to Pred(numJobs) Do
        memo1.lines.add( Format(
          'Job %s, Status (%d): %s',
          [SavePChar(pJ^[i].pDocument), pJ^[i].Status, SavePChar(pJ^[i].pStatus)] ));
  finally
    ClosePrinter( hPrinter );
  end;
end;