2DES vs. two-key triple encryption, with no decryption step
The meet-in-the-middle attack still applies; instead of attack effort $2^{57}$ DES invocations, you just increased it by $2^{56}$ extra DES block encryptions, i.e. up to about $2^{57.6}$ in total: a mere +50% increase. On the other hand, you also made the overall block cipher (your two-key triple encryption) 50% more expensive to use, so the overall security has not changed at all: attacker and defender have been slowed by the same amount.

What would make the construction much stronger is what is down with the true Triple-DES in two-key mode: $E_{k1}(D_{k2}(E_{k1}(m)))$. With the keys used alternately, the meet-in-the-middle is effectively prevented.

You could replace the middle decryption with an encryption: $E_{k1}(E_{k2}(E_{k1}(m)))$. This would still be secure, but no longer backward compatible with "plain DES", for no obvious gain (DES encryption and decryption are identical, save for the ordering of the sub-keys, so converting a decryption block to an encryption block does not seem to imply any implementation advantage).