Unable to open SSIS project in Visual Studio 2012 or 2013 (migration fails)

I am trying to open a solution file in Visual Studio 2013 Ultimate that includes an Integration Services project (.dtproj). Unfortunately, the migration failed for reasons that are slightly baffling to me.

The migration report contains the following error:

RecommenderJobs.dtproj: The application which this project type is based on was not found. Please try this link for further information: http://go.microsoft.com/fwlink/?prd=12395&pver=11&sbp=ProjectTypeDeprecated&plcid=0x409&clcid=0x409&ar=MSDN&sar=ProjectCompatibility&o1=159641D6-6404-4A2A-AE62-294DE0FE8301

Unfortunately, the link appears to be dead, so that's no help.

I also received the following error message when I tried to open this in Visual Studio 2012:

I've tried opening this solution in both Visual Studio 2012 Ultimate and Visual Studio 2013 Premium (which is why I tagged both). Both are on Update 5. I already tried updating all of my extensions and updates, even ones that seem like they wouldn't be directly relevant to the problem (like Visual Studio's Office tools).

I found a similar question on Microsoft's forum. The suggestion there was to upgrade my version of SSIS to the latest version. I tried to install SSDT for VS2013 as suggested, but the install failed:

When I checked the logs, it had the following error:

Error 0x80070666: Cannot install a product when a newer version is installed.

I do, in fact, have a different version installed, but it's not actually newer as far as I can tell (in fact, it launches in the Visual Studio 2010 shell), so I'm a little confused by this. The version information is below:

Microsoft SQL Server Integration Services Designer
Version 11.0.5058.0
Microsoft SQL Server Reporting Services Designers 
Version 11.0.5058.0

I do also have tools related to SQL Server 2012 installed.

Based on what I've read in other posts, I'd expect to see the SQL Server Data Tools in "Add/Remove Programs," but it's not. Does anyone know why that might be?

I've also seen related questions for other types of projects, such as for a PowerShell project. (Obviously, my question has nothing to do with Powershell, but the OP there was receiving the same exact error message for a different project type). Unfortunately, the suggestions on that post didn't seem all that helpful - for example, the accepted answer there involved completely uninstalling and reinstalling Visual Studio, which would be a major pain to say the least. I'm also not all that confident that that would work for me since this problem occurs in two separate versions of Visual Studio.

Does anyone have any suggestions on what else I could try?

You likely need to install both Integration services and Business Intelligence for Visual Studio (BIDS). You can do it launching SQL Server Setup and adding features to your current SQL Server Instance.

Have a look at this one: https://guruprasadv.wordpress.com/2009/06/17/opening-ssis-package-in-visual-studio/

BIDS is the IDE (integrated into Visual Studio) that allows you to drag and drop the components and create the SSIS packages. From SQL Server 2014 on wards you won't see BIDS anymore packaged in the SQL Server Setup, so you will have to install it independently. If you need it for VS 2013 here you have the link: https://www.microsoft.com/en-us/download/details.aspx?id=42313