The conditions of the Apache license are the following:
(a) You must give any other recipients of the Work or Derivative Works a copy of this License; and
(b) You must cause any modified files to carry prominent notices stating that You changed the files; and
(c) You must retain, in the Source form of any Derivative Works that You distribute, all copyright, patent, trademark, and attribution notices from the Source form of the Work, excluding those notices that do not pertain to any part of the Derivative Works; and
(d) If the Work includes a "NOTICE" text file as part of its...
If you don't intend to modify the files that you are using, you don't need to worry about clause (b).
If, as it is the case here, there is no specific copyright, patent, trademark or attribution notice, then you do not have to worry about clause (c).
If, as it is the case here, there is no NOTICE file, you don't have to worry about clause (d).
So all you have to do is keep the LICENSE file. Since you intend to release your software under Apache 2.0 anyway, you already have this license file. I would still (as suggested by Mans) add a link to the original repository, mentioning that you are depending on some of the files there (which are also licensed under Apache 2.0), wherever you plan to put your own copyright notice (for instance in the README).