No Gaps for the Letter K
The letter K has no exceptions. Every bold, italic, script, Fraktur, double-struck, sans-serif and monospace form sits in the Mathematical Alphanumeric Symbols block (U+1D400 to U+1D7FF), so a whole word set in one of those styles keeps a consistent look. Other letters are not as tidy: script B, Fraktur C and double-struck N, for example, live in an older block.
Small Capital, Turned and Other One-Off Forms
Small capital K is แด (U+1D0B). Like most small capitals in Unicode it was encoded for phonetic notation, not as a font feature, which is why small caps text pastes as ordinary characters. Upside-down text uses the turned k, ส (U+029E). Superscript k is แต (U+1D4F). Subscript k is โ (U+2096). The regional indicator ๐ฐ is half of a flag: two of them in a row turn into a flag emoji when they spell a country code.
Which K Is Safe to Paste
Bold ๐, italic, sans-serif and monospace forms display on nearly every phone and computer. Circled โ and fullwidth ๏ผซ are also widely covered. The squared and dark squared forms of K have no emoji version, so they stay black and white. Screen readers may announce a styled K by its full Unicode name or skip it, so keep anything important in plain letters as well.