That is very odd. The "Open on page" column shows your webparts are already on the page. Usually the issue is under that column they are not open, and you need to re-open (of which they show up under a special category in the webpart ribbon for closed webparts).
Check that you are not viewing the "personal view" of the page. If that is not the cause then I would disable all of those on the page and one by one start re-enabling them and testing the page output. If the first one fails - don't call it quits, disable it and try and different one first.
If you have access to the server you can try and export all your webparts and recreate the page.
Example PowerShell script courtesy Rada Tut
Add-PSSnapin Microsoft.SharePoint.PowerShell -ErrorAction SilentlyContinue
function EnsureDirectory($exportFolderPath)
{
if ( -not (Test-Path $exportFolderPath) ) {New-Item $exportFolderPath -Type Directory | Out-Null}
}
function ExportAllWebParts($siteUrl,$pageUrl,$exportFolderPath)
{
$web = Get-SPWeb $siteUrl
$wpm = $web.GetLimitedWebPartManager($pageUrl, [System.Web.UI.WebControls.WebParts.PersonalizationScope]::Shared)
EnsureDirectory $exportFolderPath
foreach($wp in $wpm.WebParts)
{
$wp.ExportMode="All";
$exportPath = $exportFolderPath + "\" + $wp.Title + ".xml"
$xwTmp = new-object System.Xml.XmlTextWriter($exportPath,$null);
$xwTmp.Formatting = 1;#Indent
$wpm.ExportWebPart($wp, $xwTmp);
$xwTmp.Flush();
$xwTmp.Close();
}
}
ExportAllWebParts $args[0] $args[1] $args[2]
# I use this script as a file, ExportWebParts.ps1. The required arguments for this script should be:
#The site absolute URL
#The page site-relative URL
#The folder path for export
#example usage ./ExportWebParts.ps1 http://spdevel.portal.com/Pages/default.aspx C:\temp\Export