Some ideas for games to play during an indoor kids' birthday party are Pin the Tail on the Donkey, Musical Chairs and Simon Says. Bingo is another good game to play during an indoor kids' birthday party.
To play Pin the Tail on the Donkey, hang a picture of a donkey without a tail on a wall. Line the party guests up a few feet away from the picture and give them each a donkey tail easily identifiable as theirs, either by a certain color or by their initials written on it. Blindfold the players, gently spin them to induce a mild state of disorientation, and turn them towards the donkey picture to attach the tail (via tape, thumbtack or fabric fastener) where they think it should go. After each child's turn, the one whose tail is closest to where it should be on the donkey picture wins the game.
Musical Chairs is another idea for an indoor kids' birthday party game. Line up two rows of chairs back to back, with one fewer chair than there are kids playing. The kids walk clockwise around the chairs while music is playing; when it stops, everybody scrambles for a seat. Eliminate the player who does not have a seat, start the music again, and remove another chair. Play continues until there is only one child left.