Why're you landing Sneak Attack with almost every strike? If you mean Summon Shadow, that is not really more powerful than a first level Druid's Animal Companion, which also provides free flanking all the time. For sneak attack, you can also just flank with a teammate. Or do you mean Hide in Plain Sight? This ability only lets you get in one sneak attack, then you'll have to hide again, which is normally a move action. Doesn't strike me as too powerful, really.
Also, this PrC only comes online at the mid levels, when sneak attack damage all the time is basically what is expected of any decent rogue-ish guy. At that point, it is also what makes such a character at all viable, compared to the Wizard throwing around Dimension Door and Black Tentacles and Lesser Planar Binding, the Cleric tanking it with persistent Divine Power, Delay Death and Beastland Ferocity, and the Barbarian one-shotting anything he can charge.
Furthermore, sneak attack really isn't that powerful even on its own. It doesn't work vs. roughly one third of all monsters out there. It doesn't work if the target has concealment. And if it does work, it's just a few d6 added to damage.
Let's review the class at 10th, 15th, and 20th level, with Assassin and Wizard casting to fulfill the spellcasting prereq. Just basic progressions, no fancy sub levels or anything.
10th:
Rogue 5/Assassin 3/This Shadowdancer 2:
Sneak Attack +6d6
Casts as a 4th level Assassin (3/1 spells/day)
BAB +6
Base Saves of +2/+10/+5
HP: 35.5 + 10*Con mod
Trapfinding, Evasion, Improved Uncanny Dodge, Trap Sense +1, Poison Use, Death Attack, Hide in Plain Sight, Darkvision
Rogue 5/Wizard 3/This Shadowdancer 2:
Sneak Attack +4d6
Casts as a 4th level Wizard (3/2 spells/day)
BAB +5
Base Saves of +2/+8/+7
HP: 32.5 + 10*Con mod
Trapfinding, Evasion, Uncanny Dodge, Trap Sense +1, Familiar, Scribe Scroll, Hide in Plain Sight, Darkvision
15th:
Rogue 5/Assassin 3/This Shadowdancer 7:
Sneak Attack +8d6
Casts as a 6th level Assassin (3/3/1 spells/day)
BAB +10
Base Saves of +4/+12/+7
HP: 48 + 10*Con mod
Trapfinding, Evasion, Improved Uncanny Dodge, Trap Sense +1, Poison Use, Death Attack, Hide in Plain Sight, Darkvision, Shadow Illusion, Summon Shadow II, Shadow Jump (40'), Defensive Roll, Slippery Mind
Rogue 5/Wizard 3/This Shadowdancer 7:
Sneak Attack +6d6
Casts as a 6th level Wizard (3/3/2 spells/day)
BAB +9
Base Saves of +4/+10/+9
HP: 45 + 10*Con mod
Trapfinding, Evasion, Improved Uncanny Dodge, Trap Sense +1, Familiar, Scribe Scroll, Hide in Plain Sight, Darkvision, Shadow Illusion, Summon Shadow II, Shadow Jump (40'), Defensive Roll, Slippery Mind
20th:
Rogue 5/Assassin 5/This Shadowdancer 10:
Sneak Attack +10d6
Casts as a 9th level Assassin (3/3/3/2 spells/day)
BAB +13
Base Saves of +5/+15/+9
HP: 62.5 + 10*Con mod
Trapfinding, Improved Evasion, Improved Uncanny Dodge, Trap Sense +1, Poison Use, Death Attack, Hide in Plain Sight, Darkvision, Shadow Illusion, Summon Shadow III, Shadow Jump (160'), Defensive Roll, Slippery Mind
Rogue 5/Wizard 5/This Shadowdancer 10:
Sneak Attack +7d6
Casts as a 9th level Wizard (4/4/3/2/1 spells/day)
BAB +12
Base Saves of +4/+12/+12
HP: 57.5 + 10*Con mod
Trapfinding, Improved Evasion, Uncanny Dodge, Trap Sense +1, Familiar, Scribe Scroll, Wizard Bonus Feat, Hide in Plain Sight, Darkvision, Shadow Illusion, Summon Shadow III, Shadow Jump (160'), Defensive Roll, Slippery Mind
As you can see (I believe), these simple, unoptimized but theoretically decent progressions make for bad 20th level characters. Even at 15th level, they seem a little lackluster to me, although the builds using Assassin do provide nice synergies and good skillmonkeying. The variant builds using Wizard are worse. In exchange for a slightly better spell list and more casting in the 20th level example, you're giving up even more BAB, quite a bit of sneak attack, and a few HP.
All right, you're doing lots of sneak attack damage, which you can deal reliably. But at 10th level, this is already declining in usefulness.
Doing +6d6 damage (i.e., circa +21) with each attack sounds great, and you get to roll a lot of dice. Two weapons will mean lots of extra damage. But remember you have to get a full attack off, and you need to connect with them. At BAB +6, this will be a problem.
The problem is only aggravated at higher levels, where your bonus damage doesn't increase very rapidly, but your BAB drops off further in comparison to a simple Barbarian. BAB +13 at 20th level is horrible for melee.
You get a lot of nice tools to play with, but to be honest, magic items or spellcasting can provide these just as well, and much more. I know this is true of all of 3.x D&D, but that doesn't make the argument invalid.
Your HP are extremely low for a melee character - ranged won't work since it's hard to get reliable sneak attacks going at range, and then only at 30'. Saves aren't great, although the Wizard based variant at least gets a decent will save.
All in all, color me underwhelmed.