Sometimes Aikido practioners tends to forget to ATEMI. Its a very powerful technique to confuse and distract the opponent.
Most likely the WT practioner you were trying to spar is highly adept in force to force confrontation. Just by waiting for him to move and look for an opening you are actually playing on his terms. Try doing some Atemi's on his face and body the moment he attacks you. Once distracted you can go around him away from his attacks and apply Aikido Techniques.
O'Sensei was once visited by an Jujitsu practioner and asked to spar with him. O'Sensei let him spar with Saito (if Iam not mistaken) Saito just used atemi's everytime the jujitsu practioner comes close to him.
BTW, I do spar and practice with several arts Tae Kwon Do, Karate and boxing. Its hard but my Aikido techniques work very well.