The Atlassian SDK is for writing apps or modules for Atlassian Server/DC products, and it's not something you'd "use in visual studio". You might use it alongside VS to do things like creating skeleton code, compiling and running the Java environments Atlassian all work in, or providing the libraries and references, after you've set up Visual Studio as a full Java development environment.
But it has noting to do with asp.net or mvc5.
You mention the REST API - you can code to talk to that in any language you want, but the SDK isn't going to help you.
For writing for the REST API, have a browse through https://developer.atlassian.com/cloud/jira/platform/rest/v3/intro/
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