How can a lattice attack be applied to ECDSA signatures?
Sure you can do. There are many lattice attacks, using your second assumption, to ECDSA (which also applied to DSA). For instance see  Smart and Howgrave-Graham and Shparlinski and Nguyen. All the lattice attacks base on finding small solutions (for the ephemeral key $k$ and the private key $a$) to the signing equation $sk-ra\equiv H(m)\pmod q.$ If you have enough signatures (i.e. enough $k_i$) then you construct a linear system $\bmod q$ and then a suitable lattice and if you have enough bits from every $k_i$ and $a$ and enough messages you can solve the CVP problem. Also you can apply Coppersmith method to the signing equation.