As an additional comment on this, note that if we define $T(n)$ for all $n$ and not just powers of two with $T(0)=0$ and $T(1)=1$ like this
$$T(n) = \sum_{k=1}^{\lfloor \log_2 n \rfloor} T(\lfloor n/2^k \rfloor)
+ n \lfloor \log_2 n \rfloor,$$
and the binary representation of $n$ is given by
$$n = \sum_{k=0}^{\lfloor \log_2 n \rfloor} d_k 2^k$$
then the exact formula for all $n$ where $n\ge 2$ is
$$T(n) = 2^{\lfloor \log_2 n \rfloor - 1} + \sum_{j=0}^{\lfloor \log_2 n \rfloor}
(\lfloor \log_2 n \rfloor -j )
[z^j] \frac{1-z}{1-2z} \sum_{k=j}^{\lfloor \log_2 n \rfloor} d_k 2^{k-j}$$
because $$\frac{1}{1-z-z^2-z^3-\cdots} = \frac{1-z}{1-2z}.$$
This simplifies to
$$2^{\lfloor \log_2 n \rfloor - 1} + \lfloor \log_2 n \rfloor \times n +
\sum_{j=1}^{\lfloor \log_2 n \rfloor}
(\lfloor \log_2 n \rfloor -j )
[z^j] \frac{1-z}{1-2z} \sum_{k=j}^{\lfloor \log_2 n \rfloor} d_k 2^{k-j}$$
Now note that for $j\ge 1$
$$[z^j] \frac{1-z}{1-2z} = 2^{j-1}$$ so this in turn again simplifies, this time to
$$2^{\lfloor \log_2 n \rfloor - 1} + \lfloor \log_2 n \rfloor \times n +
\sum_{j=1}^{\lfloor \log_2 n \rfloor}
(\lfloor \log_2 n \rfloor -j )
2^{j-1} \sum_{k=j}^{\lfloor \log_2 n \rfloor} d_k 2^{k-j}$$
which is
$$2^{\lfloor \log_2 n \rfloor - 1} + \lfloor \log_2 n \rfloor \times n +
\frac{1}{2} \sum_{j=1}^{\lfloor \log_2 n \rfloor}
(\lfloor \log_2 n \rfloor -j )
\sum_{k=j}^{\lfloor \log_2 n \rfloor} d_k 2^k.$$
Now for an upper bound consider a string of one digits giving
$$T(n)\le 2^{\lfloor \log_2 n \rfloor - 1} + \lfloor \log_2 n \rfloor \times n +
\frac{1}{2} \sum_{j=1}^{\lfloor \log_2 n \rfloor}
(\lfloor \log_2 n \rfloor -j ) (2^{\lfloor \log_2 n \rfloor+1}-2^j)
\\= 2^{\lfloor \log_2 n \rfloor - 1} + \lfloor \log_2 n \rfloor \times n +
\frac{1}{2} (\lfloor \log_2 n \rfloor +1)
\left(\lfloor \log_2 n \rfloor 2^{\lfloor \log_2 n \rfloor}
- 2^{\lfloor \log_2 n \rfloor+1} + 2\right).$$
For a lower bound consider a one digit followed by zeros giving
$$T(n)\ge 2^{\lfloor \log_2 n \rfloor - 1} + \lfloor \log_2 n \rfloor \times n +
\frac{1}{2} \sum_{j=1}^{\lfloor \log_2 n \rfloor}
(\lfloor \log_2 n \rfloor -j ) 2^{\lfloor \log_2 n \rfloor}
\\ = 2^{\lfloor \log_2 n \rfloor - 1} + \lfloor \log_2 n \rfloor \times n +
\frac{1}{4} \lfloor \log_2 n \rfloor(\lfloor \log_2 n \rfloor - 1)
2^{\lfloor \log_2 n \rfloor}.$$
The upper and lower bounds are attained and cannot be improved upon.
Selecting the dominant terms from the two bounds we finally get a complexity of
$$\Theta\left((\lfloor \log_2 n \rfloor)^2\times 2^{\lfloor \log_2 n \rfloor}\right)
= \Theta\left((\log_2 n)^2 2^{\log_2 n}\right)
= \Theta\left(n \times (\log_2 n)^2\right).$$
A similar calculation was done at this MSE link.
Addendum. The sequence of values of $T(2^j)$ using our definition is
$$3, 12, 40, 120, 336, 896, 2304, 5760, 14080, 33792,\ldots$$
which agrees (as it ought to) with the formula found by @Ragnar
$$T(2^j) = 2^{j-2}\times(2+3j+j^2).$$
(This is just a re-write of our lower bound, which to give credit was posted second.)
Remark as of Sat Feb 8 20:37:05 CET 2014. The upper limit of the outer sum can be replaced by $\lfloor \log_2 n\rfloor-1$ but the formulas are correct as stated.
