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Summary:
Add feature to Kate's static word wrap (Tools -> Word Wrap Document) that aligns the selected text vertically to the indentation level of the first line.

It appears that this feature exists for Kate's Dynamic Word Wrap. See:

Align dynamically wrapped lines to indentation depth:

Enables the start of dynamically wrapped lines to be aligned vertically to the indentation level of the first line. This can help to make code and markup more readable. http://docs.kde.org/development/en/kdesdk/kate/config-dialog-editor.html

Description:
Reformatting selected text is very simple and fast.

Under Settings -> Configure Shortcuts, I changed the shortcut for "Word Wrap Document" to "Ctrl+W".

In my documents I'll select text I want to reformat then, "Ctrl+J" to "Join Lines", and then "Ctrl+W" to "Word Wrap".

I'd like the following (everything aligned to the first line):
....Lorem ipsum dolor sit amet, consectetur adipiscing elit. Suspendisse eget
....nisl turpis. Aenean in erat massa, et imperdiet nisl. Aliquam a felis arcu,
....nec porta nunc. Nulla ac nulla et leo gravida sodales ac et magna. Nulla
....tristique tempor nulla, et sagittis mi mollis a! Cras diam ipsum, tempor
....non.

But I get (everything aligned to column 1, ignoring first line indent):
....Lorem ipsum dolor sit amet, consectetur adipiscing elit. Suspendisse eget
nisl turpis. Aenean in erat massa, et imperdiet nisl. Aliquam a felis arcu, nec
porta nunc. Nulla ac nulla et leo gravida sodales ac et magna. Nulla tristique
tempor nulla, et sagittis mi mollis a! Cras diam ipsum, tempor non.

Note: The "...." represent spaces/indents.

I manually adjust each line after the first line to line-up vertically with the first indented line.