... I don't think Glenn is right. (Yikes... I'm waiting for lightning to strike me or something. Well, I'm still here, so I'll continue.)
SP Ionic is one of the old style TrueType fonts, so if changing from Times New Roman to SP Ionic displays the characters, then it means that the text is not Unicode.
If you can change the font, for example, to SBL Greek or Cardo, and it looks good, then you are using Unicode.
To get back to your original issue, there are a variety of Unicode Greek keyboards you can use.
Check my blog post. In particular:
- I don't like the way that the Logos keyboards construct the polytonic Greek characters, but it's pretty easy to modify their existing Greek keyboard to switch letters around. (E.g., I switched ξ and χ.)
- I haven't checked closely, but the Tyndale keyboard may be fairly close to the BWGRKL keyboard.
- Tavultesoft has a number of different Greek keyboards you can try. (And if you want to spend more, you can program your own layout.)